[{"Name":"Introduction to Related Rates Problems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction, Worked Example 1","Duration":"13m 19s","ChapterTopicVideoID":1500,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/1500.jpeg","UploadDate":"2019-11-14T07:04:52.8970000","DurationForVideoObject":"PT13M19S","Description":null,"MetaTitle":"Introduction and Worked Example 1: Video + Workbook | Proprep","MetaDescription":"Related Rates Problems - Introduction to Related Rates Problems. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/calculus-i%2c-ii-and-iii/related-rates-problems/introduction-to-related-rates-problems/vid1490","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this clip, I\u0027ll be showing you how to solve"},{"Start":"00:03.570 ","End":"00:05.925","Text":"related rates word problems."},{"Start":"00:05.925 ","End":"00:08.760","Text":"This is a category of problems that\u0027s"},{"Start":"00:08.760 ","End":"00:12.315","Text":"somewhat common and it has to do with rates."},{"Start":"00:12.315 ","End":"00:16.589","Text":"A rate is a measure of how a thing is changing in time."},{"Start":"00:16.589 ","End":"00:20.940","Text":"For example, temperature could be rising"},{"Start":"00:20.940 ","End":"00:23.700","Text":"at so many degrees per minute,"},{"Start":"00:23.700 ","End":"00:25.800","Text":"or someone could be producing something"},{"Start":"00:25.800 ","End":"00:28.275","Text":"at so many units a day,"},{"Start":"00:28.275 ","End":"00:32.130","Text":"or a car could be traveling at so many kilometers"},{"Start":"00:32.130 ","End":"00:33.750","Text":"an hour and so on."},{"Start":"00:33.750 ","End":"00:36.540","Text":"When things are changing and we want to measure"},{"Start":"00:36.540 ","End":"00:38.565","Text":"the quantity per unit time,"},{"Start":"00:38.565 ","End":"00:40.625","Text":"then that\u0027s the concept of rate."},{"Start":"00:40.625 ","End":"00:43.730","Text":"In all these problems, you\u0027ll be seeing variables"},{"Start":"00:43.730 ","End":"00:45.515","Text":"that are dependent on time."},{"Start":"00:45.515 ","End":"00:47.480","Text":"The independent variable will be time,"},{"Start":"00:47.480 ","End":"00:51.079","Text":"whether it\u0027s in seconds, minutes, hours, whatever."},{"Start":"00:51.079 ","End":"00:54.200","Text":"But that\u0027s what\u0027s common to all of them and essentially,"},{"Start":"00:54.200 ","End":"00:57.380","Text":"the rate is the derivative of the quantity"},{"Start":"00:57.380 ","End":"01:00.095","Text":"that\u0027s changing with respect to time."},{"Start":"01:00.095 ","End":"01:03.110","Text":"Related rates means that in the word problem"},{"Start":"01:03.110 ","End":"01:05.330","Text":"there will be 2 or more of these variables"},{"Start":"01:05.330 ","End":"01:07.250","Text":"that are changing in time and we have"},{"Start":"01:07.250 ","End":"01:10.940","Text":"some equation or some relation between these variables."},{"Start":"01:10.940 ","End":"01:13.460","Text":"That\u0027s all I want to say at a high level."},{"Start":"01:13.460 ","End":"01:17.165","Text":"Let\u0027s jump into the first example and I\u0027ll read it."},{"Start":"01:17.165 ","End":"01:19.370","Text":"A tanker is leaking oil,"},{"Start":"01:19.370 ","End":"01:23.315","Text":"which is spreading in the shape of a circular stain."},{"Start":"01:23.315 ","End":"01:26.930","Text":"The radius of the stain is growing"},{"Start":"01:26.930 ","End":"01:30.995","Text":"at a constant rate of 0.5 meters a second."},{"Start":"01:30.995 ","End":"01:34.670","Text":"At what rate is the area of the stain growing"},{"Start":"01:34.670 ","End":"01:37.070","Text":"when the radius is 20 meters?"},{"Start":"01:37.070 ","End":"01:40.120","Text":"It\u0027s often advisable to read"},{"Start":"01:40.120 ","End":"01:42.320","Text":"the question more than once because"},{"Start":"01:42.320 ","End":"01:44.090","Text":"they\u0027re often hard to digest."},{"Start":"01:44.090 ","End":"01:46.100","Text":"Even this is a relatively simple 1"},{"Start":"01:46.100 ","End":"01:47.860","Text":"and I had to read it more than once,"},{"Start":"01:47.860 ","End":"01:49.865","Text":"1 of the things to note,"},{"Start":"01:49.865 ","End":"01:51.800","Text":"which is the most important in the"},{"Start":"01:51.800 ","End":"01:53.510","Text":"example of what the variables"},{"Start":"01:53.510 ","End":"01:57.155","Text":"are and particularly which ones are changing over time."},{"Start":"01:57.155 ","End":"02:01.180","Text":"The whole concept of rate relates to change over time."},{"Start":"02:01.180 ","End":"02:04.040","Text":"Also pay attention to which direction. I mean,"},{"Start":"02:04.040 ","End":"02:07.040","Text":"is the quantity increasing or decreasing?"},{"Start":"02:07.040 ","End":"02:09.830","Text":"That\u0027s also important. Often, the question"},{"Start":"02:09.830 ","End":"02:12.620","Text":"doesn\u0027t state things in positive and negative numbers,"},{"Start":"02:12.620 ","End":"02:14.765","Text":"but rather in increasing, decreasing,"},{"Start":"02:14.765 ","End":"02:19.520","Text":"or in this case, growing or shrinking, and so on."},{"Start":"02:19.520 ","End":"02:22.367","Text":"Here I can see 2 such variables,"},{"Start":"02:22.367 ","End":"02:25.580","Text":"1 of them is the radius of the stain,"},{"Start":"02:25.580 ","End":"02:29.810","Text":"and I\u0027d like to highlight it, and the other 1"},{"Start":"02:29.810 ","End":"02:32.730","Text":"is the area of the stain."},{"Start":"02:35.090 ","End":"02:39.020","Text":"The word related will come in when we relate"},{"Start":"02:39.020 ","End":"02:40.820","Text":"the area to the radius."},{"Start":"02:40.820 ","End":"02:44.270","Text":"Also note that this problem deals in physical quantities"},{"Start":"02:44.270 ","End":"02:45.680","Text":"and they have units."},{"Start":"02:45.680 ","End":"02:49.100","Text":"For example, here we have length,"},{"Start":"02:49.100 ","End":"02:51.965","Text":"that\u0027s radius and it\u0027s measured in meters,"},{"Start":"02:51.965 ","End":"02:56.060","Text":"and we have a rate of growth which is in meters a second."},{"Start":"02:56.060 ","End":"02:58.850","Text":"For 1 thing that tells me that the unit of time"},{"Start":"02:58.850 ","End":"03:01.265","Text":"is probably going to be seconds."},{"Start":"03:01.265 ","End":"03:05.360","Text":"Now I\u0027d like to talk about a strategy for the solution."},{"Start":"03:05.360 ","End":"03:08.780","Text":"I\u0027ve done several of these problems and they all seem"},{"Start":"03:08.780 ","End":"03:11.525","Text":"to follow the same general steps."},{"Start":"03:11.525 ","End":"03:13.340","Text":"Each 1 is different, of course,"},{"Start":"03:13.340 ","End":"03:15.590","Text":"but there\u0027s a general flow of things."},{"Start":"03:15.590 ","End":"03:18.830","Text":"Over time I\u0027ve made myself a little checklist that I put at"},{"Start":"03:18.830 ","End":"03:22.730","Text":"the side and I was deliberating whether to share it with you."},{"Start":"03:22.730 ","End":"03:24.590","Text":"You don\u0027t have to follow it in detail,"},{"Start":"03:24.590 ","End":"03:27.530","Text":"but I\u0027ve presented it here just so you can see that I\u0027m following"},{"Start":"03:27.530 ","End":"03:32.105","Text":"a certain checklist with a mnemonic to jog my memory."},{"Start":"03:32.105 ","End":"03:35.780","Text":"We\u0027ll expand on these as we solve the problem."},{"Start":"03:35.780 ","End":"03:40.190","Text":"But just roughly, the first step is where I give variables names"},{"Start":"03:40.190 ","End":"03:42.650","Text":"and I call that denote and often that goes"},{"Start":"03:42.650 ","End":"03:45.140","Text":"with a sketch and then they\u0027re together."},{"Start":"03:45.140 ","End":"03:48.290","Text":"Then I deduce from these variables,"},{"Start":"03:48.290 ","End":"03:51.020","Text":"the derivatives of these variables."},{"Start":"03:51.020 ","End":"03:54.740","Text":"Then I state what I\u0027m given in the question"},{"Start":"03:54.740 ","End":"03:56.795","Text":"and what I\u0027m looking for in the answer."},{"Start":"03:56.795 ","End":"03:59.635","Text":"Then I find an equation that ties them all,"},{"Start":"03:59.635 ","End":"04:01.760","Text":"ties some of these quantities together"},{"Start":"04:01.760 ","End":"04:04.460","Text":"and I take the derivative of that equation also."},{"Start":"04:04.460 ","End":"04:06.830","Text":"I substitute known quantities,"},{"Start":"04:06.830 ","End":"04:08.660","Text":"I solve, and finally,"},{"Start":"04:08.660 ","End":"04:11.090","Text":"I write a sentence in plain English to interpret"},{"Start":"04:11.090 ","End":"04:14.900","Text":"the solution in human language."},{"Start":"04:14.900 ","End":"04:17.585","Text":"That\u0027s just to show you that there is a general method."},{"Start":"04:17.585 ","End":"04:20.865","Text":"But forget that for now."},{"Start":"04:20.865 ","End":"04:24.665","Text":"We\u0027ll start with the first step."},{"Start":"04:24.665 ","End":"04:28.385","Text":"Now call that denote,"},{"Start":"04:28.385 ","End":"04:32.610","Text":"where I denote certain quantities with names."},{"Start":"04:32.750 ","End":"04:36.950","Text":"There\u0027s 1 thing that is not commonly done and that is"},{"Start":"04:36.950 ","End":"04:40.220","Text":"to mention in the list of variables, time itself."},{"Start":"04:40.220 ","End":"04:41.765","Text":"But I\u0027ll do it,"},{"Start":"04:41.765 ","End":"04:49.315","Text":"this one\u0027s and say that t is time and it also has certain units."},{"Start":"04:49.315 ","End":"04:51.490","Text":"Sometimes the question has mixed units,"},{"Start":"04:51.490 ","End":"04:53.470","Text":"seconds and minutes all mixed together,"},{"Start":"04:53.470 ","End":"04:54.940","Text":"you have to decide what your unit is."},{"Start":"04:54.940 ","End":"04:56.590","Text":"We\u0027re going to go with seconds."},{"Start":"04:56.590 ","End":"04:58.849","Text":"That\u0027s the only 1 that\u0027s mentioned."},{"Start":"04:58.849 ","End":"05:02.200","Text":"The next variable that changes with time"},{"Start":"05:02.200 ","End":"05:04.315","Text":"is this 1 I\u0027ve highlighted,"},{"Start":"05:04.315 ","End":"05:07.585","Text":"and it\u0027s a radius so let\u0027s give it the letter r. Now,"},{"Start":"05:07.585 ","End":"05:09.790","Text":"r is not just a variable,"},{"Start":"05:09.790 ","End":"05:12.415","Text":"it\u0027s a function of time, actually."},{"Start":"05:12.415 ","End":"05:15.370","Text":"It changes over time."},{"Start":"05:15.370 ","End":"05:20.560","Text":"So r of t is the radius of"},{"Start":"05:20.560 ","End":"05:27.460","Text":"the stain and it\u0027s a unit of length"},{"Start":"05:27.460 ","End":"05:30.050","Text":"and it\u0027s measured in meters."},{"Start":"05:30.770 ","End":"05:33.790","Text":"The other variable that changes over time"},{"Start":"05:33.790 ","End":"05:35.500","Text":"is the area of the stain,"},{"Start":"05:35.500 ","End":"05:37.315","Text":"let\u0027s call it A."},{"Start":"05:37.315 ","End":"05:45.790","Text":"Also as a function of time and that\u0027s the area of the stain."},{"Start":"05:46.250 ","End":"05:48.510","Text":"What would area be?"},{"Start":"05:48.510 ","End":"05:50.350","Text":"If length is in meters,"},{"Start":"05:50.350 ","End":"05:54.145","Text":"area would be in square meters, meter squared."},{"Start":"05:54.145 ","End":"05:57.100","Text":"Next on my list is something cryptic,"},{"Start":"05:57.100 ","End":"06:00.320","Text":"which I call, hence."},{"Start":"06:00.680 ","End":"06:05.365","Text":"In English word means from here or follows from,"},{"Start":"06:05.365 ","End":"06:08.890","Text":"that r prime of t,"},{"Start":"06:08.890 ","End":"06:11.650","Text":"this is the part where I write the derivatives,"},{"Start":"06:11.650 ","End":"06:14.240","Text":"is the rate of change of the radius."},{"Start":"06:14.240 ","End":"06:18.830","Text":"Notice that the derivative when the independent variable is time,"},{"Start":"06:18.830 ","End":"06:22.280","Text":"is rate of change, change over time."},{"Start":"06:22.280 ","End":"06:27.375","Text":"Rate of change of the radius and"},{"Start":"06:27.375 ","End":"06:33.980","Text":"A prime also of t is the rate of change of the area."},{"Start":"06:33.980 ","End":"06:39.125","Text":"Now both of these have units and the rule is"},{"Start":"06:39.125 ","End":"06:45.410","Text":"that whenever you have a quantity that\u0027s measured in 1 unit,"},{"Start":"06:45.410 ","End":"06:50.780","Text":"the rate of change has the same unit per unit time."},{"Start":"06:50.780 ","End":"06:52.249","Text":"That sounds complicated."},{"Start":"06:52.249 ","End":"06:55.279","Text":"What I mean is that if radius is in meters,"},{"Start":"06:55.279 ","End":"06:59.165","Text":"then the rate of change is in meters per second."},{"Start":"06:59.165 ","End":"07:02.945","Text":"If area is in square meters,"},{"Start":"07:02.945 ","End":"07:07.520","Text":"then rate of change of area is in square meters per second,"},{"Start":"07:07.520 ","End":"07:09.455","Text":"that\u0027s our unit of time."},{"Start":"07:09.455 ","End":"07:12.589","Text":"My next heading, I call it given,"},{"Start":"07:12.589 ","End":"07:16.805","Text":"which is just what I\u0027m given in the problem."},{"Start":"07:16.805 ","End":"07:25.610","Text":"What we\u0027re given is in the question, here it is."},{"Start":"07:25.610 ","End":"07:29.090","Text":"The radius is growing at a constant rate"},{"Start":"07:29.090 ","End":"07:32.925","Text":"of 0.5 meters a second."},{"Start":"07:32.925 ","End":"07:36.575","Text":"Radius is growing means the rate of change of the radius."},{"Start":"07:36.575 ","End":"07:41.180","Text":"Rate of change of the radius is 0.5"},{"Start":"07:41.180 ","End":"07:42.530","Text":"and because its rate of change,"},{"Start":"07:42.530 ","End":"07:45.760","Text":"it\u0027s not meters but meters per second,"},{"Start":"07:45.760 ","End":"07:49.810","Text":"the original quantity per unit of time."},{"Start":"07:50.080 ","End":"07:55.505","Text":"That\u0027s the given part and now next is what we\u0027re looking for."},{"Start":"07:55.505 ","End":"07:59.300","Text":"What we\u0027re looking for is written here in the question part."},{"Start":"07:59.300 ","End":"08:03.320","Text":"At what rate is the area of the stain growing"},{"Start":"08:03.320 ","End":"08:05.460","Text":"when the radius is 20 meters?"},{"Start":"08:05.460 ","End":"08:08.480","Text":"What we need to know is the growing"},{"Start":"08:08.480 ","End":"08:11.870","Text":"is to do with the changing and has to do with the rate of change."},{"Start":"08:11.870 ","End":"08:19.335","Text":"What we\u0027re looking for is A prime of t,"},{"Start":"08:19.335 ","End":"08:21.070","Text":"but we don\u0027t know what t is,"},{"Start":"08:21.070 ","End":"08:30.170","Text":"but we do know that we\u0027re asking for the time when r equals 20."},{"Start":"08:30.170 ","End":"08:33.610","Text":"We\u0027re looking for A prime at a specific moment in time,"},{"Start":"08:33.610 ","End":"08:35.845","Text":"but it\u0027s not defined as a time unit,"},{"Start":"08:35.845 ","End":"08:40.700","Text":"but as an event which is when the radius is 20."},{"Start":"08:41.660 ","End":"08:46.545","Text":"The next stage is to find,"},{"Start":"08:46.545 ","End":"08:48.940","Text":"I\u0027m looking at my checklist, that\u0027s just for me."},{"Start":"08:48.940 ","End":"08:51.980","Text":"I\u0027m looking for an equation."},{"Start":"08:52.910 ","End":"08:57.790","Text":"Well, 1 or more equations that tie in some"},{"Start":"08:57.790 ","End":"09:00.615","Text":"of these variables together."},{"Start":"09:00.615 ","End":"09:06.084","Text":"In our case, the equation that I\u0027m going to use,"},{"Start":"09:06.084 ","End":"09:09.090","Text":"I hope you remember your geometry,"},{"Start":"09:09.090 ","End":"09:10.340","Text":"if not I\u0027ll remind you,"},{"Start":"09:10.340 ","End":"09:13.625","Text":"we need the area of a circle given its radius."},{"Start":"09:13.625 ","End":"09:17.960","Text":"In general, we can recite A equals Pi r squared,"},{"Start":"09:17.960 ","End":"09:23.175","Text":"but in our case, there\u0027s a matter of time involved."},{"Start":"09:23.175 ","End":"09:26.760","Text":"What I get is that A, but it\u0027s not A,"},{"Start":"09:26.760 ","End":"09:30.840","Text":"it\u0027s A of t equals Pi r squared."},{"Start":"09:30.840 ","End":"09:35.520","Text":"Pi is Pi, r is r of t squared,"},{"Start":"09:35.520 ","End":"09:39.000","Text":"I\u0027d better put brackets around it"},{"Start":"09:39.000 ","End":"09:41.865","Text":"and that\u0027s going to be my main equation."},{"Start":"09:41.865 ","End":"09:45.130","Text":"When we have this equation,"},{"Start":"09:46.370 ","End":"09:49.755","Text":"I better make a general comment."},{"Start":"09:49.755 ","End":"09:54.005","Text":"When you\u0027re given a homework or an exam,"},{"Start":"09:54.005 ","End":"09:57.680","Text":"you\u0027re not expected to write these headings. If you do,"},{"Start":"09:57.680 ","End":"09:59.855","Text":"then make them more meaningful,"},{"Start":"09:59.855 ","End":"10:02.330","Text":"I find they\u0027re a good idea for solving the problem,"},{"Start":"10:02.330 ","End":"10:03.620","Text":"and I\u0027ve included them here,"},{"Start":"10:03.620 ","End":"10:07.775","Text":"but certainly not mandatory to use these."},{"Start":"10:07.775 ","End":"10:09.740","Text":"These are just suggestions and I would"},{"Start":"10:09.740 ","End":"10:13.200","Text":"afterwards leave them out or just do them in my head."},{"Start":"10:14.620 ","End":"10:19.595","Text":"I\u0027m just following the steps that I wrote for myself and next,"},{"Start":"10:19.595 ","End":"10:24.210","Text":"it says to me derive and what does derive mean? Yes, I remember."},{"Start":"10:24.740 ","End":"10:27.050","Text":"If you derive this equation,"},{"Start":"10:27.050 ","End":"10:29.870","Text":"you\u0027ll get more information on how the rates"},{"Start":"10:29.870 ","End":"10:32.970","Text":"of change are tied in."},{"Start":"10:33.410 ","End":"10:36.500","Text":"In the derived step,"},{"Start":"10:36.500 ","End":"10:40.130","Text":"I just differentiate from A of t,"},{"Start":"10:40.130 ","End":"10:46.725","Text":"I get A prime of t. Here I used the chain rule,"},{"Start":"10:46.725 ","End":"10:49.089","Text":"Pi stays in front,"},{"Start":"10:49.089 ","End":"10:56.880","Text":"from r squared we get 2r and don\u0027t forget,"},{"Start":"10:56.880 ","End":"10:58.080","Text":"r is not just r,"},{"Start":"10:58.080 ","End":"11:00.335","Text":"it\u0027s a function of t so we have to,"},{"Start":"11:00.335 ","End":"11:03.080","Text":"because of the chain rule, put the internal derivative,"},{"Start":"11:03.080 ","End":"11:08.040","Text":"times r prime of t. Here I have 1 equation that relates"},{"Start":"11:08.040 ","End":"11:09.980","Text":"A and r but what I really wanted was"},{"Start":"11:09.980 ","End":"11:13.010","Text":"this 1 that ties in the rates of change,"},{"Start":"11:13.010 ","End":"11:15.760","Text":"A prime is related to r prime."},{"Start":"11:15.760 ","End":"11:18.540","Text":"Next on my checklist is substitute"},{"Start":"11:18.540 ","End":"11:21.870","Text":"and what I substitute relates to what I\u0027m"},{"Start":"11:21.870 ","End":"11:28.350","Text":"looking for and I\u0027m looking for A prime when r equals 20."},{"Start":"11:28.350 ","End":"11:32.360","Text":"What I\u0027m going to substitute is that r equals"},{"Start":"11:32.360 ","End":"11:37.430","Text":"20 here and the other thing I know is that r prime,"},{"Start":"11:37.430 ","End":"11:40.685","Text":"this 1 we\u0027re given, that r prime is 0.5."},{"Start":"11:40.685 ","End":"11:45.260","Text":"Let\u0027s write that A prime of t equals 2Pi,"},{"Start":"11:45.260 ","End":"11:51.210","Text":"then r of t is the 20 because that\u0027s the time we\u0027re asked about."},{"Start":"11:51.460 ","End":"11:54.500","Text":"Then r prime of t,"},{"Start":"11:54.500 ","End":"11:58.065","Text":"which is constantly 0.5."},{"Start":"11:58.065 ","End":"12:05.120","Text":"Times 0.5 and well,"},{"Start":"12:05.120 ","End":"12:06.965","Text":"I have a section that says solve,"},{"Start":"12:06.965 ","End":"12:09.750","Text":"but there\u0027s nothing much to solve here"},{"Start":"12:09.750 ","End":"12:12.795","Text":"so I\u0027ll just continue with this,"},{"Start":"12:12.795 ","End":"12:14.540","Text":"usually it\u0027s more complicated,"},{"Start":"12:14.540 ","End":"12:16.370","Text":"so there\u0027s a whole stage of solving."},{"Start":"12:16.370 ","End":"12:18.470","Text":"But here we can just say that it\u0027s equal"},{"Start":"12:18.470 ","End":"12:21.320","Text":"to 20 times 0.5 is 10,"},{"Start":"12:21.320 ","End":"12:23.330","Text":"10 times 2 is 20."},{"Start":"12:23.330 ","End":"12:26.160","Text":"So it\u0027s 20 Pi."},{"Start":"12:26.650 ","End":"12:30.170","Text":"It\u0027s not mandatory, but it\u0027s quite customary"},{"Start":"12:30.170 ","End":"12:33.110","Text":"to write the final answer as a decimal."},{"Start":"12:33.110 ","End":"12:35.885","Text":"This is approximately equal to,"},{"Start":"12:35.885 ","End":"12:41.970","Text":"according to my calculator 62.83."},{"Start":"12:42.320 ","End":"12:46.430","Text":"Finally, I\u0027ll write something humanly readable"},{"Start":"12:46.430 ","End":"12:48.455","Text":"and then I\u0027ll also put the units in"},{"Start":"12:48.455 ","End":"12:51.170","Text":"and this like how we used to give our answers in"},{"Start":"12:51.170 ","End":"12:54.605","Text":"grade school and we give a proper sentence like,"},{"Start":"12:54.605 ","End":"12:59.570","Text":"when the radius is 20 meters,"},{"Start":"12:59.570 ","End":"13:06.385","Text":"the area is growing at a rate of 62.83 meters a second."},{"Start":"13:06.385 ","End":"13:11.030","Text":"What I\u0027d like to do now is to present you"},{"Start":"13:11.030 ","End":"13:13.910","Text":"with a new problem to solve,"},{"Start":"13:13.910 ","End":"13:16.550","Text":"from this time we\u0027ll be more familiar with the steps,"},{"Start":"13:16.550 ","End":"13:19.290","Text":"possibly 1 with a sketch."}],"ID":1490},{"Watched":false,"Name":"Worked Example 2","Duration":"12m 34s","ChapterTopicVideoID":1501,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.655","Text":"Here\u0027s a second example."},{"Start":"00:02.655 ","End":"00:06.960","Text":"A rocket rises vertically from its launchpad"},{"Start":"00:06.960 ","End":"00:11.160","Text":"at a constant speed of 260 meters a second."},{"Start":"00:11.160 ","End":"00:17.385","Text":"A camera positioned 900 meters away tracks its ascent."},{"Start":"00:17.385 ","End":"00:21.090","Text":"When the rocket is 1,200 meters up,"},{"Start":"00:21.090 ","End":"00:26.960","Text":"2 questions: A, at what rate is the angle of elevation"},{"Start":"00:26.960 ","End":"00:29.480","Text":"of the camera increasing?"},{"Start":"00:29.480 ","End":"00:33.420","Text":"They want the answer in degrees per second."},{"Start":"00:33.440 ","End":"00:37.775","Text":"I hope you remember how to convert radians to degrees."},{"Start":"00:37.775 ","End":"00:42.200","Text":"B, at what rate is the distance between"},{"Start":"00:42.200 ","End":"00:45.095","Text":"the rocket and the camera increasing?"},{"Start":"00:45.095 ","End":"00:48.820","Text":"Note that they use the word increasing and not decreasing."},{"Start":"00:48.820 ","End":"00:50.930","Text":"If you think about it, it has to be increasing."},{"Start":"00:50.930 ","End":"00:53.675","Text":"But if we get a negative answer for the rate,"},{"Start":"00:53.675 ","End":"00:56.330","Text":"then we\u0027ll know that we did something wrong."},{"Start":"00:56.330 ","End":"00:58.430","Text":"I\u0027ll explain all the terms if you don\u0027t know"},{"Start":"00:58.430 ","End":"01:01.580","Text":"what angle of elevation is and so on."},{"Start":"01:01.580 ","End":"01:05.250","Text":"Whenever possible, begin with a sketch."},{"Start":"01:05.470 ","End":"01:08.980","Text":"There\u0027s the sketch."},{"Start":"01:08.980 ","End":"01:12.555","Text":"I\u0027m doing the sketch and the note at the same time."},{"Start":"01:12.555 ","End":"01:17.540","Text":"In the diagram, all the objects I\u0027ve named"},{"Start":"01:17.540 ","End":"01:19.900","Text":"and the 2 constants that were given,"},{"Start":"01:19.900 ","End":"01:23.230","Text":"the rate at which the rocket is going up"},{"Start":"01:23.230 ","End":"01:25.840","Text":"and the distance from the camera to the pad,"},{"Start":"01:25.840 ","End":"01:28.630","Text":"what we\u0027re missing are the variable names,"},{"Start":"01:28.630 ","End":"01:30.565","Text":"the ones that are changing with time,"},{"Start":"01:30.565 ","End":"01:33.610","Text":"and 2 of them are written here,"},{"Start":"01:33.610 ","End":"01:36.100","Text":"angle of elevation which is this,"},{"Start":"01:36.100 ","End":"01:38.995","Text":"I\u0027ll call this Alpha,"},{"Start":"01:38.995 ","End":"01:41.155","Text":"that\u0027s for an angle,"},{"Start":"01:41.155 ","End":"01:45.090","Text":"and distance between the rocket and the camera."},{"Start":"01:45.090 ","End":"01:47.895","Text":"I\u0027ll call it d for distance."},{"Start":"01:47.895 ","End":"01:50.240","Text":"There\u0027s 1 that\u0027s not exactly mentioned,"},{"Start":"01:50.240 ","End":"01:52.760","Text":"but it\u0027s hinted that the height of the rocket."},{"Start":"01:52.760 ","End":"01:55.310","Text":"But somewhere in here, and I\u0027ll call that h,"},{"Start":"01:55.310 ","End":"01:57.830","Text":"just because of the phrasing, we missed it."},{"Start":"01:57.830 ","End":"02:00.020","Text":"All of these are of course a functions of t"},{"Start":"02:00.020 ","End":"02:02.420","Text":"so what I\u0027m going to write is I\u0027m going to"},{"Start":"02:02.420 ","End":"02:06.995","Text":"write Alpha of t for the angle of elevation."},{"Start":"02:06.995 ","End":"02:10.820","Text":"We have h of t is the height and d is the distance."},{"Start":"02:10.820 ","End":"02:16.910","Text":"Here, we have angle of elevation, and here,"},{"Start":"02:16.910 ","End":"02:20.825","Text":"we have the height of the rocket, and finally,"},{"Start":"02:20.825 ","End":"02:25.910","Text":"our d is the distance from rocket to camera."},{"Start":"02:25.910 ","End":"02:29.990","Text":"Next, we do the derivatives of these,"},{"Start":"02:29.990 ","End":"02:32.675","Text":"and that gives us extra information."},{"Start":"02:32.675 ","End":"02:35.810","Text":"Hence, we\u0027re basically giving descriptions"},{"Start":"02:35.810 ","End":"02:38.150","Text":"of the derivatives of all these quantities,"},{"Start":"02:38.150 ","End":"02:39.560","Text":"of Alpha prime of t,"},{"Start":"02:39.560 ","End":"02:47.310","Text":"of h prime of t, and d prime of t."},{"Start":"02:47.310 ","End":"02:50.120","Text":"Each of these is the rate of change of the corresponding 1 here,"},{"Start":"02:50.120 ","End":"02:52.330","Text":"I\u0027ll write a few words."},{"Start":"02:52.330 ","End":"02:55.880","Text":"The rate of change of the angle of elevation,"},{"Start":"02:55.880 ","End":"02:58.925","Text":"rate of change of the height of the rocket,"},{"Start":"02:58.925 ","End":"03:01.850","Text":"the rate of change of the distance between"},{"Start":"03:01.850 ","End":"03:04.345","Text":"the camera and the rocket."},{"Start":"03:04.345 ","End":"03:07.010","Text":"I forgot to write the units,"},{"Start":"03:07.010 ","End":"03:08.870","Text":"but it\u0027s not too late."},{"Start":"03:08.870 ","End":"03:14.935","Text":"Angle of elevation, we said we\u0027re going to take that in radians,"},{"Start":"03:14.935 ","End":"03:17.670","Text":"and work all the way through with radians."},{"Start":"03:17.670 ","End":"03:20.900","Text":"Finally, at the end, if they want something in degrees,"},{"Start":"03:20.900 ","End":"03:22.580","Text":"we\u0027ll convert at the end."},{"Start":"03:22.580 ","End":"03:26.290","Text":"The height, well, we\u0027re only working in meters here."},{"Start":"03:26.290 ","End":"03:27.870","Text":"So that\u0027s in meters,"},{"Start":"03:27.870 ","End":"03:28.950","Text":"the height of the rocket."},{"Start":"03:28.950 ","End":"03:31.290","Text":"The distance from the rocket to the camera,"},{"Start":"03:31.290 ","End":"03:34.020","Text":"that\u0027s also in meters."},{"Start":"03:34.020 ","End":"03:36.530","Text":"The rates of change are the same as these,"},{"Start":"03:36.530 ","End":"03:39.300","Text":"except per unit time which was going to be the second,"},{"Start":"03:39.300 ","End":"03:41.795","Text":"so t is going to be time in seconds."},{"Start":"03:41.795 ","End":"03:43.520","Text":"Rate of change of angle,"},{"Start":"03:43.520 ","End":"03:47.445","Text":"and that\u0027s in radians per second."},{"Start":"03:47.445 ","End":"03:52.610","Text":"Rate of change of height is in meters per second."},{"Start":"03:52.610 ","End":"03:54.920","Text":"Just want to mention that when it\u0027s a"},{"Start":"03:54.920 ","End":"03:56.510","Text":"rate of change of some distance,"},{"Start":"03:56.510 ","End":"03:59.915","Text":"a unit of length, it\u0027s often considered as the speed."},{"Start":"03:59.915 ","End":"04:02.450","Text":"An actual fact, the rate of change of the height of"},{"Start":"04:02.450 ","End":"04:05.940","Text":"the rocket is just the speed of the rocket."},{"Start":"04:06.140 ","End":"04:10.455","Text":"The same, rate of change of distance, meters a second."},{"Start":"04:10.455 ","End":"04:13.895","Text":"That\u0027s that part. Following this,"},{"Start":"04:13.895 ","End":"04:16.280","Text":"we have to know what it is that we\u0027re given"},{"Start":"04:16.280 ","End":"04:18.275","Text":"and what it is that we\u0027re looking for."},{"Start":"04:18.275 ","End":"04:19.900","Text":"I scrolled back up,"},{"Start":"04:19.900 ","End":"04:25.605","Text":"and we can see that what we\u0027re looking for is here,"},{"Start":"04:25.605 ","End":"04:30.890","Text":"that the rate of change of the angle of elevation,"},{"Start":"04:30.890 ","End":"04:33.830","Text":"and the rate of change of the distance"},{"Start":"04:33.830 ","End":"04:35.540","Text":"between the rocket and camera,"},{"Start":"04:35.540 ","End":"04:36.775","Text":"and both of these,"},{"Start":"04:36.775 ","End":"04:41.290","Text":"when the rocket is 1,200 meters up."},{"Start":"04:43.760 ","End":"04:48.980","Text":"We\u0027re given that it\u0027s launched and it goes up"},{"Start":"04:48.980 ","End":"04:52.850","Text":"at a constant speed of 260 meters a second."},{"Start":"04:52.850 ","End":"04:55.250","Text":"Let\u0027s write all these things down there."},{"Start":"04:55.250 ","End":"04:59.554","Text":"The 260 meters a second is under the given,"},{"Start":"04:59.554 ","End":"05:02.085","Text":"and that\u0027s the h part here,"},{"Start":"05:02.085 ","End":"05:09.695","Text":"so that we\u0027re given that h prime of t is equal to 260."},{"Start":"05:09.695 ","End":"05:16.010","Text":"What we\u0027re looking for is that when h equals 1,200,"},{"Start":"05:16.010 ","End":"05:17.900","Text":"which I copied from there,"},{"Start":"05:17.900 ","End":"05:25.560","Text":"what are Alpha prime and h prime?"},{"Start":"05:25.560 ","End":"05:27.990","Text":"2 different things we have to find"},{"Start":"05:27.990 ","End":"05:32.115","Text":"but the same instant when h is 1,200."},{"Start":"05:32.115 ","End":"05:38.675","Text":"Let\u0027s begin with a. I repeated part of the picture over here."},{"Start":"05:38.675 ","End":"05:44.060","Text":"Now, the equation for a which will connect Alpha"},{"Start":"05:44.060 ","End":"05:47.885","Text":"and h is gotten from trigonometry."},{"Start":"05:47.885 ","End":"05:50.045","Text":"We have a right-angled triangle,"},{"Start":"05:50.045 ","End":"05:52.710","Text":"and we have this angle here,"},{"Start":"05:52.710 ","End":"05:56.565","Text":"and we can just use the tangent."},{"Start":"05:56.565 ","End":"06:01.995","Text":"If we write the tangent of Alpha,"},{"Start":"06:01.995 ","End":"06:06.960","Text":"it\u0027s the side opposite which is h over"},{"Start":"06:06.960 ","End":"06:10.095","Text":"the adjacent side which is 900."},{"Start":"06:10.095 ","End":"06:11.630","Text":"I was going to do b later."},{"Start":"06:11.630 ","End":"06:12.910","Text":"It was so close to everything,"},{"Start":"06:12.910 ","End":"06:15.280","Text":"so let\u0027s write equation b also."},{"Start":"06:15.280 ","End":"06:17.795","Text":"For this, I\u0027ll just use Pythagoras\u0027 theorem."},{"Start":"06:17.795 ","End":"06:21.350","Text":"Meanwhile, we\u0027ll continue with a and leave b alone."},{"Start":"06:21.350 ","End":"06:26.240","Text":"On my checklist, after equation, I have derived."},{"Start":"06:26.240 ","End":"06:29.105","Text":"Let\u0027s differentiate what it says here."},{"Start":"06:29.105 ","End":"06:31.100","Text":"To differentiate, you have to remember"},{"Start":"06:31.100 ","End":"06:34.380","Text":"a little bit of the trigonometric derivatives,"},{"Start":"06:34.380 ","End":"06:37.790","Text":"or you can just take my word for it that a derivative of"},{"Start":"06:37.790 ","End":"06:41.565","Text":"tangent is 1 over cosine squared,"},{"Start":"06:41.565 ","End":"06:47.969","Text":"so I\u0027ll write it as 1 over cosine squared Alpha,"},{"Start":"06:47.969 ","End":"06:53.010","Text":"but we have to also multiply the 1"},{"Start":"06:53.010 ","End":"06:56.250","Text":"over cosine squared by Alpha prime of t"},{"Start":"06:56.250 ","End":"06:58.665","Text":"which I\u0027ll put in the numerator,"},{"Start":"06:58.665 ","End":"07:01.010","Text":"internal derivative, so to speak."},{"Start":"07:01.010 ","End":"07:02.845","Text":"This is going to equal,"},{"Start":"07:02.845 ","End":"07:05.025","Text":"h prime of t,"},{"Start":"07:05.025 ","End":"07:08.175","Text":"the 900 is a constant so it just stays."},{"Start":"07:08.175 ","End":"07:12.045","Text":"After derive, I usually have substitute,"},{"Start":"07:12.045 ","End":"07:15.570","Text":"and then I go to substitute and I realize that we\u0027re missing"},{"Start":"07:15.570 ","End":"07:19.550","Text":"something because we don\u0027t have the cosine of Alpha,"},{"Start":"07:19.550 ","End":"07:21.305","Text":"but we could compute it."},{"Start":"07:21.305 ","End":"07:24.290","Text":"Because you see, we\u0027re taking a snapshot"},{"Start":"07:24.290 ","End":"07:28.090","Text":"at a certain moment in time when h is 1,200."},{"Start":"07:28.090 ","End":"07:29.910","Text":"When h is 1,200,"},{"Start":"07:29.910 ","End":"07:32.575","Text":"I have 2 sides of the right-angled triangle,"},{"Start":"07:32.575 ","End":"07:36.705","Text":"and I could compute the hypotenuse using Pythagoras\u0027 theorem."},{"Start":"07:36.705 ","End":"07:41.900","Text":"I\u0027ll spare you the details but we actually"},{"Start":"07:41.900 ","End":"07:47.385","Text":"get that d is equal to 1,500,"},{"Start":"07:47.385 ","End":"07:49.960","Text":"not d that we want, it\u0027s cosine Alpha."},{"Start":"07:49.960 ","End":"07:54.080","Text":"Cosine is the adjacent side over the hypotenuse."},{"Start":"07:54.080 ","End":"08:03.925","Text":"We get that the cosine of Alpha is 900 over 1,500."},{"Start":"08:03.925 ","End":"08:06.220","Text":"I can write this as 3 over 5,"},{"Start":"08:06.220 ","End":"08:09.680","Text":"for those who like decimals, 0.6, whatever."},{"Start":"08:09.680 ","End":"08:12.470","Text":"We\u0027re ready to substitute because Alpha prime"},{"Start":"08:12.470 ","End":"08:14.015","Text":"is what we\u0027re looking for."},{"Start":"08:14.015 ","End":"08:23.300","Text":"We get Alpha prime of t over 3/5 squared is equal to now,"},{"Start":"08:23.300 ","End":"08:24.590","Text":"h prime t. We have it,"},{"Start":"08:24.590 ","End":"08:26.255","Text":"it\u0027s even on the page here,"},{"Start":"08:26.255 ","End":"08:29.225","Text":"it\u0027s 260 at this point in time,"},{"Start":"08:29.225 ","End":"08:32.820","Text":"and 900 doesn\u0027t change."},{"Start":"08:33.050 ","End":"08:36.135","Text":"Just a little bit of mathematics,"},{"Start":"08:36.135 ","End":"08:38.860","Text":"and we come to the answer that"},{"Start":"08:38.860 ","End":"08:42.580","Text":"Alpha prime of t at that particular point in"},{"Start":"08:42.580 ","End":"08:50.535","Text":"time is 0.104 radians per second, that\u0027s the unit."},{"Start":"08:50.535 ","End":"08:54.090","Text":"The question did ask for in degrees,"},{"Start":"08:54.090 ","End":"08:56.920","Text":"and I\u0027m not going to do that just to give you"},{"Start":"08:56.920 ","End":"08:58.285","Text":"instructions how to do it."},{"Start":"08:58.285 ","End":"09:01.825","Text":"To go from radians to degrees,"},{"Start":"09:01.825 ","End":"09:06.700","Text":"you just multiply by 180 over Pi."},{"Start":"09:06.700 ","End":"09:09.905","Text":"You have calculators, I\u0027m not going to pursue that."},{"Start":"09:09.905 ","End":"09:16.995","Text":"Then what we do to finish with part a is to just interpret,"},{"Start":"09:16.995 ","End":"09:20.915","Text":"and this just means write it in plain English,"},{"Start":"09:20.915 ","End":"09:25.085","Text":"something like when the rocket is 1,200 meters up,"},{"Start":"09:25.085 ","End":"09:28.880","Text":"the angle of elevation is increasing at a rate of"},{"Start":"09:28.880 ","End":"09:33.680","Text":"5.96 degrees a second, something like that."},{"Start":"09:33.680 ","End":"09:37.880","Text":"I think we can go on and do part b. I copied"},{"Start":"09:37.880 ","End":"09:41.090","Text":"all the important information for part b"},{"Start":"09:41.090 ","End":"09:43.099","Text":"from what we did before."},{"Start":"09:43.099 ","End":"09:47.090","Text":"The equation that ties in the variables"},{"Start":"09:47.090 ","End":"09:49.540","Text":"is this 1 from Pythagoras."},{"Start":"09:49.540 ","End":"09:56.445","Text":"Basically says that d squared equals h squared plus 900 squared."},{"Start":"09:56.445 ","End":"09:58.700","Text":"I copied the little sketch."},{"Start":"09:58.700 ","End":"10:02.780","Text":"I made a note that at that point in time when h is 1,200,"},{"Start":"10:02.780 ","End":"10:05.635","Text":"that d we computed at 1,500."},{"Start":"10:05.635 ","End":"10:09.845","Text":"Also, the given was that h prime,"},{"Start":"10:09.845 ","End":"10:13.490","Text":"that\u0027s the speed of the rocket upwards, is 260."},{"Start":"10:13.490 ","End":"10:16.640","Text":"What we\u0027re looking for is d prime,"},{"Start":"10:16.640 ","End":"10:19.400","Text":"the rate of change of the distance between"},{"Start":"10:19.400 ","End":"10:23.040","Text":"the camera and the rocket."},{"Start":"10:24.140 ","End":"10:28.380","Text":"The next heading, so to speak,"},{"Start":"10:28.380 ","End":"10:31.430","Text":"after equation, I call derive or we take"},{"Start":"10:31.430 ","End":"10:34.460","Text":"the derivative of this equation."},{"Start":"10:34.460 ","End":"10:45.750","Text":"We get 2d of t but times internal derivative d prime of t,"},{"Start":"10:45.750 ","End":"10:49.275","Text":"equals, same thing with h,"},{"Start":"10:49.275 ","End":"10:54.500","Text":"2h of t h prime of t, it\u0027s the chain rule."},{"Start":"10:54.500 ","End":"10:56.390","Text":"This isn\u0027t goes to nothing."},{"Start":"10:56.390 ","End":"11:00.865","Text":"I can immediately can cancel the 2 with the 2."},{"Start":"11:00.865 ","End":"11:03.450","Text":"Let\u0027s see where we stand."},{"Start":"11:03.450 ","End":"11:07.680","Text":"H prime, we have, that\u0027s here, h,"},{"Start":"11:07.680 ","End":"11:10.700","Text":"we have, d, we have,"},{"Start":"11:10.700 ","End":"11:13.580","Text":"so we got all we need to compute d prime."},{"Start":"11:13.580 ","End":"11:17.240","Text":"Let\u0027s just substitute and also solve"},{"Start":"11:17.240 ","End":"11:22.000","Text":"in the same short calculations."},{"Start":"11:22.000 ","End":"11:30.090","Text":"d is 1,500,"},{"Start":"11:30.090 ","End":"11:33.350","Text":"d prime is what we\u0027re looking for."},{"Start":"11:33.350 ","End":"11:44.345","Text":"h is 1,200 and h prime is 260."},{"Start":"11:44.345 ","End":"11:50.745","Text":"From all of this, we get that d prime is equal to"},{"Start":"11:50.745 ","End":"11:59.000","Text":"1,200 times 260 over 1,500 which equals,"},{"Start":"11:59.000 ","End":"12:04.585","Text":"with the help of a calculator, 208."},{"Start":"12:04.585 ","End":"12:09.780","Text":"Finally, just to write something humanly readable."},{"Start":"12:09.920 ","End":"12:13.820","Text":"Here\u0027s an example of what I mean by humanly readable."},{"Start":"12:13.820 ","End":"12:19.920","Text":"I\u0027ll take this 208 and answer it properly like in school."},{"Start":"12:19.920 ","End":"12:22.790","Text":"When the rocket is 1,200 meters up,"},{"Start":"12:22.790 ","End":"12:25.550","Text":"its distance from the camera is increasing"},{"Start":"12:25.550 ","End":"12:29.180","Text":"at a rate of 208 meters per second."},{"Start":"12:29.180 ","End":"12:30.380","Text":"This is what we were looking for"},{"Start":"12:30.380 ","End":"12:34.170","Text":"but in a nice English sentence. We\u0027re done."}],"ID":1491},{"Watched":false,"Name":"Exercise 1","Duration":"6m 11s","ChapterTopicVideoID":6282,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have to determine the rate"},{"Start":"00:03.570 ","End":"00:04.830","Text":"at which the radius of"},{"Start":"00:04.830 ","End":"00:08.160","Text":"a spherical balloon is increasing when"},{"Start":"00:08.160 ","End":"00:11.760","Text":"the diameter of the balloon is 40 centimeters,"},{"Start":"00:11.760 ","End":"00:14.970","Text":"if it is known that air is being pumped into"},{"Start":"00:14.970 ","End":"00:19.530","Text":"the balloon at a rate of 10 cubic centimeters a minute."},{"Start":"00:19.530 ","End":"00:21.210","Text":"Let me rephrase."},{"Start":"00:21.210 ","End":"00:26.160","Text":"We\u0027re given a balloon which is being inflated at a constant rate,"},{"Start":"00:26.160 ","End":"00:27.360","Text":"and this is the rate,"},{"Start":"00:27.360 ","End":"00:29.970","Text":"so the balloon is getting larger."},{"Start":"00:29.970 ","End":"00:32.204","Text":"As the balloon grows,"},{"Start":"00:32.204 ","End":"00:36.195","Text":"its volume grows, but so does its radius."},{"Start":"00:36.195 ","End":"00:40.380","Text":"What we want to know is when the balloon is a certain size,"},{"Start":"00:40.380 ","End":"00:44.360","Text":"what rate does the radius change?"},{"Start":"00:44.360 ","End":"00:46.010","Text":"Usually we draw a little diagram,"},{"Start":"00:46.010 ","End":"00:48.155","Text":"but this is not complicated enough."},{"Start":"00:48.155 ","End":"00:51.230","Text":"First thing we do in most of these exercises"},{"Start":"00:51.230 ","End":"00:54.210","Text":"is to name some variables,"},{"Start":"00:54.210 ","End":"00:56.090","Text":"you could call them, but really I\u0027m looking"},{"Start":"00:56.090 ","End":"00:59.540","Text":"for functions which are changing over time."},{"Start":"00:59.540 ","End":"01:03.995","Text":"What I\u0027m going to do is to denote which means to name."},{"Start":"01:03.995 ","End":"01:07.985","Text":"First quantity we have is the volume"},{"Start":"01:07.985 ","End":"01:10.430","Text":"because we\u0027re pumping air in and this is"},{"Start":"01:10.430 ","End":"01:12.425","Text":"a unit of volume, so we have,"},{"Start":"01:12.425 ","End":"01:15.800","Text":"let\u0027s call it V. V will be a function of"},{"Start":"01:15.800 ","End":"01:19.985","Text":"time and the volume is the volume of the balloon,"},{"Start":"01:19.985 ","End":"01:22.560","Text":"that\u0027s what I\u0027m interested in."},{"Start":"01:22.670 ","End":"01:26.220","Text":"The other quantity is the radius."},{"Start":"01:26.220 ","End":"01:27.630","Text":"Let\u0027s call that R of t,"},{"Start":"01:27.630 ","End":"01:31.405","Text":"which is the radius, also of the balloon."},{"Start":"01:31.405 ","End":"01:35.764","Text":"These are physical quantities and we should give them units."},{"Start":"01:35.764 ","End":"01:40.695","Text":"The volume is measured in centimeters cubed,"},{"Start":"01:40.695 ","End":"01:45.000","Text":"cubic centimeters and we can see that here"},{"Start":"01:45.000 ","End":"01:47.185","Text":"and the radius being a unit of length,"},{"Start":"01:47.185 ","End":"01:49.880","Text":"will be just in centimeters."},{"Start":"01:49.880 ","End":"01:54.150","Text":"The exercise we\u0027re talking about the diameter also."},{"Start":"01:54.150 ","End":"01:55.800","Text":"Somewhere there, we should remember"},{"Start":"01:55.800 ","End":"01:58.330","Text":"that diameter is twice the radius."},{"Start":"01:58.330 ","End":"02:01.190","Text":"These are the basic quantities that change over time."},{"Start":"02:01.190 ","End":"02:04.580","Text":"We can talk about their rates of change, hence,"},{"Start":"02:04.580 ","End":"02:09.235","Text":"we have that V prime of t,"},{"Start":"02:09.235 ","End":"02:12.120","Text":"derivative is rate of change."},{"Start":"02:12.120 ","End":"02:15.695","Text":"This is the rate of change of the volume,"},{"Start":"02:15.695 ","End":"02:17.870","Text":"the volume of the balloon of course,"},{"Start":"02:17.870 ","End":"02:23.930","Text":"the units of that will be in centimeters cubed per minute."},{"Start":"02:23.930 ","End":"02:26.450","Text":"Which means that in this exercise,"},{"Start":"02:26.450 ","End":"02:28.955","Text":"we\u0027re measuring time in minutes."},{"Start":"02:28.955 ","End":"02:33.725","Text":"Likewise, rate of change of the radius, it\u0027s also growing,"},{"Start":"02:33.725 ","End":"02:38.310","Text":"is rate of change of the radius"},{"Start":"02:38.310 ","End":"02:41.960","Text":"and that will be in centimeters a minute."},{"Start":"02:41.960 ","End":"02:44.990","Text":"Next, I want to know what I\u0027m given."},{"Start":"02:44.990 ","End":"02:49.220","Text":"What we\u0027re given is that air is being pumped"},{"Start":"02:49.220 ","End":"02:52.145","Text":"into the balloon at a rate of"},{"Start":"02:52.145 ","End":"02:54.620","Text":"10 centimeters cubed per minute"},{"Start":"02:54.620 ","End":"03:00.445","Text":"and I can write this as V prime of t is equal to 10."},{"Start":"03:00.445 ","End":"03:02.060","Text":"That\u0027s this 10 here,"},{"Start":"03:02.060 ","End":"03:05.600","Text":"the volume is increasing because the rate of change is positive."},{"Start":"03:05.600 ","End":"03:08.360","Text":"After I know what\u0027s given,"},{"Start":"03:08.360 ","End":"03:11.665","Text":"I want to know what it is that I\u0027m looking for."},{"Start":"03:11.665 ","End":"03:14.570","Text":"What I\u0027m looking for is the rate"},{"Start":"03:14.570 ","End":"03:16.970","Text":"at which the radius of the balloon is"},{"Start":"03:16.970 ","End":"03:22.540","Text":"increasing when the diameter of the balloon is 40 centimeters."},{"Start":"03:22.540 ","End":"03:25.455","Text":"I\u0027ll write this heading as looking for,"},{"Start":"03:25.455 ","End":"03:30.650","Text":"and what I\u0027m looking for is R prime of t,"},{"Start":"03:30.650 ","End":"03:33.890","Text":"which is the rate of change of the radius"},{"Start":"03:33.890 ","End":"03:36.830","Text":"when the diameter is 40."},{"Start":"03:36.830 ","End":"03:42.094","Text":"Which means that when R of t is equal to 20,"},{"Start":"03:42.094 ","End":"03:45.005","Text":"because the radius is just half the diameter."},{"Start":"03:45.005 ","End":"03:47.360","Text":"Next, we\u0027re going to need an equation,"},{"Start":"03:47.360 ","End":"03:50.675","Text":"one or more to tie in all of these variables."},{"Start":"03:50.675 ","End":"03:54.185","Text":"We can get our equation from geometry."},{"Start":"03:54.185 ","End":"03:57.699","Text":"There is a formula for the volume of a sphere,"},{"Start":"03:57.699 ","End":"04:02.760","Text":"and that is that V equals 4/3 Pi R cubed."},{"Start":"04:02.760 ","End":"04:10.004","Text":"But here, V is V of t is 4/3 Pi R,"},{"Start":"04:10.004 ","End":"04:13.765","Text":"again, R is R of t, cubed."},{"Start":"04:13.765 ","End":"04:18.260","Text":"However, what we really want is not an equation tying V and R,"},{"Start":"04:18.260 ","End":"04:20.780","Text":"but more, we want their derivatives."},{"Start":"04:20.780 ","End":"04:25.910","Text":"Let\u0027s derive this and we get that the derivative of the volume"},{"Start":"04:25.910 ","End":"04:31.675","Text":"is 4/3 times 3 is 4 Pi."},{"Start":"04:31.675 ","End":"04:33.810","Text":"Here, it\u0027s a chain rule,"},{"Start":"04:33.810 ","End":"04:40.530","Text":"so we get 3 R of t squared times"},{"Start":"04:40.530 ","End":"04:46.130","Text":"the internal derivative R prime of t. Altogether"},{"Start":"04:46.130 ","End":"04:52.090","Text":"it\u0027s 12 Pi times R of t squared,"},{"Start":"04:52.090 ","End":"04:54.650","Text":"R prime of t. Next,"},{"Start":"04:54.650 ","End":"04:56.960","Text":"we\u0027re going to substitute some of the variables,"},{"Start":"04:56.960 ","End":"04:58.610","Text":"the values that we know."},{"Start":"04:58.610 ","End":"05:02.565","Text":"We were given that V prime of t is always 10,"},{"Start":"05:02.565 ","End":"05:05.025","Text":"so 10 is equal to,"},{"Start":"05:05.025 ","End":"05:09.359","Text":"now the other thing we know is that R is 20,"},{"Start":"05:09.359 ","End":"05:12.525","Text":"or at least we\u0027re asked to find R prime when R is 20."},{"Start":"05:12.525 ","End":"05:17.190","Text":"We get 12 Pi times"},{"Start":"05:17.190 ","End":"05:22.850","Text":"20 squared R prime of t. Next,"},{"Start":"05:22.850 ","End":"05:25.040","Text":"we solve this equation for what we want,"},{"Start":"05:25.040 ","End":"05:30.005","Text":"which is R prime of t. This equals,"},{"Start":"05:30.005 ","End":"05:34.265","Text":"if you divide 10 by 12 Pi times 20 squared,"},{"Start":"05:34.265 ","End":"05:35.480","Text":"you\u0027ll do the arithmetic,"},{"Start":"05:35.480 ","End":"05:44.564","Text":"we get R prime is equal to 1/480 Pi,"},{"Start":"05:44.564 ","End":"05:47.430","Text":"when I divided 10 by this."},{"Start":"05:47.430 ","End":"05:52.190","Text":"Then finally, we just write this answer in words"},{"Start":"05:52.190 ","End":"05:54.505","Text":"and using the proper units."},{"Start":"05:54.505 ","End":"05:58.040","Text":"When the diameter is 20 centimeters,"},{"Start":"05:58.040 ","End":"06:00.200","Text":"the radius is increasing,"},{"Start":"06:00.200 ","End":"06:02.060","Text":"I like to say increasing or decreasing"},{"Start":"06:02.060 ","End":"06:04.010","Text":"depending on positive or negative,"},{"Start":"06:04.010 ","End":"06:09.670","Text":"at a rate of 140 Pi and this is in centimeters per second."},{"Start":"06:09.670 ","End":"06:12.670","Text":"That\u0027s the answer, we\u0027re done."}],"ID":6294},{"Watched":false,"Name":"Exercise 2","Duration":"7m 10s","ChapterTopicVideoID":6283,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"This is an exercise in related rates."},{"Start":"00:04.170 ","End":"00:07.485","Text":"A 20 foot ladder is resting against the wall."},{"Start":"00:07.485 ","End":"00:11.265","Text":"The bottom is initially 15 feet away from the wall,"},{"Start":"00:11.265 ","End":"00:14.190","Text":"and it\u0027s being pushed towards the wall"},{"Start":"00:14.190 ","End":"00:16.874","Text":"at a rate of 1/2 a foot per second."},{"Start":"00:16.874 ","End":"00:20.370","Text":"How fast is the top of the ladder moving up the wall,"},{"Start":"00:20.370 ","End":"00:24.180","Text":"16 seconds after we start pushing?"},{"Start":"00:24.180 ","End":"00:29.010","Text":"The first step would be to denote some variables with names."},{"Start":"00:29.010 ","End":"00:32.640","Text":"I don\u0027t see how we can do it without a sketch."},{"Start":"00:32.640 ","End":"00:37.639","Text":"Here\u0027s the sketch and let\u0027s denote the distance"},{"Start":"00:37.639 ","End":"00:43.130","Text":"from the wall to the base of the ladder by x."},{"Start":"00:43.130 ","End":"00:51.030","Text":"We\u0027ll denote the distance from the top of the ladder to the ground as y."},{"Start":"00:51.030 ","End":"00:54.110","Text":"Here\u0027s the ladder which is 20 foot-long."},{"Start":"00:54.110 ","End":"00:56.330","Text":"Here\u0027s the wall, this is the ground,"},{"Start":"00:56.330 ","End":"01:02.419","Text":"and even drawn the rates of change of x and y and these are actually speeds."},{"Start":"01:02.419 ","End":"01:04.750","Text":"Rate of change of distance is speed."},{"Start":"01:04.750 ","End":"01:06.965","Text":"X is a function of t,"},{"Start":"01:06.965 ","End":"01:10.175","Text":"and this is the distance of the foot of the ladder from the wall."},{"Start":"01:10.175 ","End":"01:11.810","Text":"It\u0027s also measured in feet,"},{"Start":"01:11.810 ","End":"01:14.400","Text":"also will let y,"},{"Start":"01:14.400 ","End":"01:17.690","Text":"which is a function of t also denote the distance of the top of"},{"Start":"01:17.690 ","End":"01:21.110","Text":"the ladder from the ground also distance measured in feet."},{"Start":"01:21.110 ","End":"01:22.565","Text":"What we really want,"},{"Start":"01:22.565 ","End":"01:23.990","Text":"or we also want,"},{"Start":"01:23.990 ","End":"01:29.405","Text":"are the derivatives of these to get from distance to speeds and hence,"},{"Start":"01:29.405 ","End":"01:34.655","Text":"x prime will be the speed of the foot of the ladder away from the wall."},{"Start":"01:34.655 ","End":"01:38.915","Text":"That\u0027s the speed this is going in this direction and"},{"Start":"01:38.915 ","End":"01:43.880","Text":"y prime denotes the speed of the top of the ladder up the wall,"},{"Start":"01:43.880 ","End":"01:46.325","Text":"measured in feet per second, both of them."},{"Start":"01:46.325 ","End":"01:48.155","Text":"Next, let\u0027s see what we are given."},{"Start":"01:48.155 ","End":"01:55.204","Text":"Well, we\u0027re given that x at time 0 is 15 feet."},{"Start":"01:55.204 ","End":"02:04.465","Text":"That\u0027s the initial and x prime at anytime t is equal to 1/2 a foot a second."},{"Start":"02:04.465 ","End":"02:07.880","Text":"But it was towards the wall,"},{"Start":"02:07.880 ","End":"02:15.005","Text":"so this is minus 1/2 and the next section we need is what we\u0027re looking for."},{"Start":"02:15.005 ","End":"02:19.160","Text":"Actually, if you think about it for y prime of 16,"},{"Start":"02:19.160 ","End":"02:22.670","Text":"which is how fast is the top of the ladder moving up the wall"},{"Start":"02:22.670 ","End":"02:28.055","Text":"16 seconds after we start pushing so this is what we\u0027re looking for."},{"Start":"02:28.055 ","End":"02:32.570","Text":"Next, we need an equation to tie up some of these variables."},{"Start":"02:32.570 ","End":"02:37.205","Text":"What I\u0027m going to write here is directly Pythagoras\u0027 theorem."},{"Start":"02:37.205 ","End":"02:40.625","Text":"We have the x here, y here,"},{"Start":"02:40.625 ","End":"02:44.600","Text":"20 here for the length of the ladder and Pythagoras\u0027 theorem,"},{"Start":"02:44.600 ","End":"02:46.475","Text":"I mean, obviously this is a right angle."},{"Start":"02:46.475 ","End":"02:48.200","Text":"The wall is usually straight."},{"Start":"02:48.200 ","End":"02:56.835","Text":"I\u0027ll go down and write the equation as x of t squared,"},{"Start":"02:56.835 ","End":"03:03.500","Text":"plus y of t squared and these vary according to time,"},{"Start":"03:03.500 ","End":"03:07.540","Text":"but the ladder doesn\u0027t change according to time is 20 squared."},{"Start":"03:07.540 ","End":"03:17.105","Text":"We get another equation from differentiating this equation to x of t. This is chain rule."},{"Start":"03:17.105 ","End":"03:23.630","Text":"X prime of t plus 2y of t,"},{"Start":"03:23.630 ","End":"03:28.670","Text":"y prime of t is equal to 0."},{"Start":"03:28.670 ","End":"03:30.215","Text":"This is a constant."},{"Start":"03:30.215 ","End":"03:34.880","Text":"The next step is to substitute some of the known quantities."},{"Start":"03:34.880 ","End":"03:41.480","Text":"Now what we\u0027re concerned with is what happens at the time when T equals 16,"},{"Start":"03:41.480 ","End":"03:48.185","Text":"because we\u0027re looking for the speed at which the ladder moves when y equals 16."},{"Start":"03:48.185 ","End":"03:49.595","Text":"Let\u0027s see what we know."},{"Start":"03:49.595 ","End":"03:51.685","Text":"We have here 2."},{"Start":"03:51.685 ","End":"03:56.810","Text":"Now we need x of 16 that we\u0027re substituting when t equals 16,"},{"Start":"03:56.810 ","End":"03:58.820","Text":"then we need x of t,"},{"Start":"03:58.820 ","End":"04:06.365","Text":"which is x of 16 is 1 of those time and distance and speed problems to time 0,"},{"Start":"04:06.365 ","End":"04:10.430","Text":"we know that x of 0 is 15."},{"Start":"04:10.430 ","End":"04:13.310","Text":"In other words, the foot of the ladder is"},{"Start":"04:13.310 ","End":"04:17.315","Text":"15 feet away from the wall and we know its speed."},{"Start":"04:17.315 ","End":"04:23.510","Text":"Its speed at every t is minus 1/2."},{"Start":"04:23.510 ","End":"04:30.140","Text":"After 16 seconds, we\u0027re going to get the equation that x of 16 is the beginning x,"},{"Start":"04:30.140 ","End":"04:32.975","Text":"which is 15, and then plus,"},{"Start":"04:32.975 ","End":"04:37.595","Text":"but in this case it\u0027s minus the speed times the time."},{"Start":"04:37.595 ","End":"04:41.090","Text":"You see if I start 15 feet away from the wall,"},{"Start":"04:41.090 ","End":"04:47.180","Text":"and I\u0027m approaching the wall at a 1/2 a foot per second and after 16 seconds,"},{"Start":"04:47.180 ","End":"04:48.830","Text":"I\u0027ll be at this position,"},{"Start":"04:48.830 ","End":"04:53.120","Text":"which is equal to 15 minus 8, which is 7."},{"Start":"04:53.120 ","End":"04:57.545","Text":"Here we have x of t is 7, so 2 times 7."},{"Start":"04:57.545 ","End":"05:01.880","Text":"Next thing, x prime of t. That\u0027s the same for"},{"Start":"05:01.880 ","End":"05:06.050","Text":"all t. That\u0027s the speed of x and we know that it\u0027s minus a 1/2."},{"Start":"05:06.050 ","End":"05:09.275","Text":"We\u0027re going towards the wall plus 2,"},{"Start":"05:09.275 ","End":"05:12.740","Text":"and then we need y of 16."},{"Start":"05:12.740 ","End":"05:16.055","Text":"This Pythagoras rule, if we look at it here,"},{"Start":"05:16.055 ","End":"05:23.090","Text":"gives us a tie in between x and y at any given t. When t is 16,"},{"Start":"05:23.090 ","End":"05:24.965","Text":"I can plug that in here,"},{"Start":"05:24.965 ","End":"05:30.935","Text":"and say that x of 16 squared plus y of 16"},{"Start":"05:30.935 ","End":"05:37.295","Text":"squared is equal to 20 squared is just Pythagoras with the ladder at any given time,"},{"Start":"05:37.295 ","End":"05:38.540","Text":"the x part squared,"},{"Start":"05:38.540 ","End":"05:42.270","Text":"and the y part squared gives us the ladder squared."},{"Start":"05:42.270 ","End":"05:46.865","Text":"We know this x of 16 because we just figured it here is 7."},{"Start":"05:46.865 ","End":"05:54.770","Text":"We get 7 squared plus y of 16 squared equals 400."},{"Start":"05:54.770 ","End":"06:00.140","Text":"We know from this that y times 16 is equal to"},{"Start":"06:00.140 ","End":"06:07.370","Text":"400 minus 49 is 351 and we need the square root of that."},{"Start":"06:07.370 ","End":"06:12.990","Text":"We can put this in here 2 times the square root of"},{"Start":"06:12.990 ","End":"06:22.315","Text":"351 and then y prime of 16 hold this equals 0."},{"Start":"06:22.315 ","End":"06:26.345","Text":"What we get if we isolate y prime of 16,"},{"Start":"06:26.345 ","End":"06:29.450","Text":"I usually put a heading solve before we get to"},{"Start":"06:29.450 ","End":"06:33.875","Text":"the actual solving as opposed to the substitute although they really go together."},{"Start":"06:33.875 ","End":"06:42.320","Text":"We get that y prime of 16 is equal to 7,"},{"Start":"06:42.320 ","End":"06:49.570","Text":"and then divide by twice the square root of 351."},{"Start":"06:49.570 ","End":"06:56.060","Text":"The last thing we do is just write it in words and throw in the units after 16 seconds,"},{"Start":"06:56.060 ","End":"06:59.720","Text":"the top of the ladder is moving upward at a rate or at"},{"Start":"06:59.720 ","End":"07:04.295","Text":"a speed of 7 over twice square root of 351."},{"Start":"07:04.295 ","End":"07:11.310","Text":"Use your calculators to decimalize this and the unit feet per second. We\u0027re done."}],"ID":6295},{"Watched":false,"Name":"Exercise 3","Duration":"6m 43s","ChapterTopicVideoID":6284,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.335","Text":"In this exercise, 2 girls are 40 feet apart,"},{"Start":"00:04.335 ","End":"00:10.050","Text":"1 of them starts walking north at a rate so that the angle shown in the diagram below"},{"Start":"00:10.050 ","End":"00:16.185","Text":"is changing at a constant rate of 0.02 radians a minute."},{"Start":"00:16.185 ","End":"00:23.365","Text":"At what rate is the distance between the 2 girls changing when Theta equals 0.5 radians?"},{"Start":"00:23.365 ","End":"00:27.605","Text":"Let\u0027s reread this but interpreted in the context of the diagram."},{"Start":"00:27.605 ","End":"00:29.629","Text":"2 girls are 40 feet apart,"},{"Start":"00:29.629 ","End":"00:31.775","Text":"1 girl is located here,"},{"Start":"00:31.775 ","End":"00:34.475","Text":"and 1 girl is located here,"},{"Start":"00:34.475 ","End":"00:39.260","Text":"and this 1 is stationary and this 1 is walking north."},{"Start":"00:39.260 ","End":"00:42.470","Text":"Here\u0027s an arrow and she\u0027s walking upward at a rate,"},{"Start":"00:42.470 ","End":"00:43.910","Text":"we don\u0027t know what that rate is,"},{"Start":"00:43.910 ","End":"00:51.965","Text":"but we know it\u0027s a rate such that the angle Theta is changing at 0.02 radians a minute."},{"Start":"00:51.965 ","End":"00:56.915","Text":"Notice that Theta is increasing as the girl walks north,"},{"Start":"00:56.915 ","End":"00:59.269","Text":"and so Theta prime is positive."},{"Start":"00:59.269 ","End":"01:02.195","Text":"Now the question is at what rate is the distance between"},{"Start":"01:02.195 ","End":"01:05.680","Text":"2 girls changing when Theta is 0.5."},{"Start":"01:05.680 ","End":"01:08.345","Text":"When Theta reaches this angle,"},{"Start":"01:08.345 ","End":"01:10.385","Text":"this girl is moving north."},{"Start":"01:10.385 ","End":"01:12.065","Text":"So the distance x,"},{"Start":"01:12.065 ","End":"01:14.930","Text":"we\u0027ll call it between the 2 girls is actually increasing and"},{"Start":"01:14.930 ","End":"01:18.875","Text":"we need to know the rate of increase of x."},{"Start":"01:18.875 ","End":"01:21.870","Text":"The first section heading is denote."},{"Start":"01:21.870 ","End":"01:26.150","Text":"When I say denote, I mean giving variables names and by variables,"},{"Start":"01:26.150 ","End":"01:29.890","Text":"I generally mean quantities that are changing over time."},{"Start":"01:29.890 ","End":"01:33.095","Text":"The variables already given names in the diagram."},{"Start":"01:33.095 ","End":"01:37.669","Text":"x but it\u0027s x of t because it\u0027s changing over time,"},{"Start":"01:37.669 ","End":"01:41.615","Text":"is equal to distance between the 2 girls."},{"Start":"01:41.615 ","End":"01:48.754","Text":"Theta, which is also changing over time as a function of t. I\u0027ll just write as shown."},{"Start":"01:48.754 ","End":"01:54.559","Text":"But Theta is growing in this direction and so x will be increasing actually,"},{"Start":"01:54.559 ","End":"01:56.885","Text":"as this girl goes north,"},{"Start":"01:56.885 ","End":"01:58.160","Text":"distance will be increasing."},{"Start":"01:58.160 ","End":"02:01.565","Text":"Next, we always derive these quantities."},{"Start":"02:01.565 ","End":"02:08.090","Text":"So we get that x prime of t is the rate at which x is increasing and"},{"Start":"02:08.090 ","End":"02:10.850","Text":"Theta prime of t is just the derivative of"},{"Start":"02:10.850 ","End":"02:14.550","Text":"Theta is the rate at which Theta is increasing,"},{"Start":"02:14.550 ","End":"02:16.215","Text":"where Theta is as in the picture."},{"Start":"02:16.215 ","End":"02:21.830","Text":"The next 2 headings will be what we\u0027re given and what we\u0027re looking for."},{"Start":"02:21.830 ","End":"02:26.885","Text":"What we\u0027re given is that Theta prime of t,"},{"Start":"02:26.885 ","End":"02:33.155","Text":"because this is changing at a constant rate, is 0.02."},{"Start":"02:33.155 ","End":"02:35.195","Text":"There\u0027s something I forgot to do."},{"Start":"02:35.195 ","End":"02:36.770","Text":"When I write these quantities,"},{"Start":"02:36.770 ","End":"02:38.500","Text":"I should really write the units."},{"Start":"02:38.500 ","End":"02:40.295","Text":"Let me correct that."},{"Start":"02:40.295 ","End":"02:42.155","Text":"Here I\u0027ve put in some units,"},{"Start":"02:42.155 ","End":"02:46.640","Text":"the distance between the girls is in feet and the angle"},{"Start":"02:46.640 ","End":"02:52.115","Text":"Theta is in radians and it looks like the unit of time is minutes,"},{"Start":"02:52.115 ","End":"02:57.770","Text":"because we\u0027re given that x prime is in feet per minute or it will be,"},{"Start":"02:57.770 ","End":"03:00.295","Text":"everything talks about minutes."},{"Start":"03:00.295 ","End":"03:02.974","Text":"Theta prime, which is what we\u0027re given actually,"},{"Start":"03:02.974 ","End":"03:05.810","Text":"is the rate at which Theta is increasing."},{"Start":"03:05.810 ","End":"03:10.490","Text":"We\u0027re told it\u0027s 0.02 radians a minute and that just takes us into this part,"},{"Start":"03:10.490 ","End":"03:13.790","Text":"which is given that Theta prime is 0.02."},{"Start":"03:13.790 ","End":"03:17.855","Text":"Length in feet, angle in radians, time in minutes."},{"Start":"03:17.855 ","End":"03:21.980","Text":"What we\u0027re looking for is the rate at which the distance is increasing."},{"Start":"03:21.980 ","End":"03:25.760","Text":"In other words, we\u0027re looking for x prime of t."},{"Start":"03:25.760 ","End":"03:31.295","Text":"But we\u0027re given a condition when Theta is 0.5 radians,"},{"Start":"03:31.295 ","End":"03:35.615","Text":"when Theta is 0.5."},{"Start":"03:35.615 ","End":"03:41.285","Text":"The next section is an equation which ties in some of these variables."},{"Start":"03:41.285 ","End":"03:46.100","Text":"Here I see I have a right angle triangle and what I can say is"},{"Start":"03:46.100 ","End":"03:51.290","Text":"that cosine of Theta is 40 over x."},{"Start":"03:51.290 ","End":"03:52.520","Text":"So here we are."},{"Start":"03:52.520 ","End":"03:55.122","Text":"Cosine Theta is 40 over x."},{"Start":"03:55.122 ","End":"03:58.175","Text":"We can also write it if we take the inverse,"},{"Start":"03:58.175 ","End":"03:59.690","Text":"the reciprocal of the cosine,"},{"Start":"03:59.690 ","End":"04:01.355","Text":"which is the secant,"},{"Start":"04:01.355 ","End":"04:05.975","Text":"We can also write secant Theta is x over 40."},{"Start":"04:05.975 ","End":"04:11.360","Text":"Actually, I prefer this form because I\u0027ve already done this exercise."},{"Start":"04:11.360 ","End":"04:14.600","Text":"I tried it both ways and this 1 comes out much"},{"Start":"04:14.600 ","End":"04:18.005","Text":"simpler because after we differentiate this,"},{"Start":"04:18.005 ","End":"04:20.120","Text":"we totally get rid of x."},{"Start":"04:20.120 ","End":"04:21.320","Text":"We only have x prime,"},{"Start":"04:21.320 ","End":"04:23.570","Text":"whereas this will get x and x prime,"},{"Start":"04:23.570 ","End":"04:25.385","Text":"and it\u0027s more complicated."},{"Start":"04:25.385 ","End":"04:29.345","Text":"Let\u0027s stay with this as our equation."},{"Start":"04:29.345 ","End":"04:32.160","Text":"Next we do the differentiation."},{"Start":"04:32.160 ","End":"04:34.450","Text":"We derive working with this 1."},{"Start":"04:34.450 ","End":"04:38.420","Text":"The derivative of secant is secant times tangent."},{"Start":"04:38.420 ","End":"04:43.940","Text":"So we get secant Theta times tangent Theta."},{"Start":"04:43.940 ","End":"04:46.460","Text":"But we also need the internal derivative"},{"Start":"04:46.460 ","End":"04:51.470","Text":"Theta prime of t is equal to set of x."},{"Start":"04:51.470 ","End":"04:54.515","Text":"We get x prime over 40."},{"Start":"04:54.515 ","End":"04:58.490","Text":"I\u0027ve been omitting the parentheses t because we know that"},{"Start":"04:58.490 ","End":"05:00.670","Text":"these things are dependent on t."},{"Start":"05:00.670 ","End":"05:02.420","Text":"After the derive,"},{"Start":"05:02.420 ","End":"05:05.825","Text":"we get a substitute section heading."},{"Start":"05:05.825 ","End":"05:09.640","Text":"What we\u0027ll substitute is when Theta is 0.5"},{"Start":"05:09.640 ","End":"05:11.900","Text":"because that\u0027s when we need x prime,"},{"Start":"05:11.900 ","End":"05:17.850","Text":"we\u0027ll get secant of 0.5 because that\u0027s what Theta is,"},{"Start":"05:17.850 ","End":"05:27.785","Text":"tangent of 0.5 and Theta prime is always 0.02."},{"Start":"05:27.785 ","End":"05:30.054","Text":"It\u0027s the constant rate in fact."},{"Start":"05:30.054 ","End":"05:34.400","Text":"That equals x prime is what we\u0027re looking for."},{"Start":"05:34.400 ","End":"05:38.120","Text":"So I\u0027ll leave that as it is, and over 40."},{"Start":"05:38.120 ","End":"05:40.685","Text":"The next section is to solve."},{"Start":"05:40.685 ","End":"05:50.855","Text":"So we get that x prime of t is equal to 40 times 0.02 is 0.8."},{"Start":"05:50.855 ","End":"05:53.120","Text":"I just have to do the rest on the calculator,"},{"Start":"05:53.120 ","End":"05:58.470","Text":"just leave it as secant 0.5, Tangent 0.5,"},{"Start":"05:58.470 ","End":"06:05.245","Text":"and 0.498, I make it approximately."},{"Start":"06:05.245 ","End":"06:11.614","Text":"The final step is to just write this in words and put in the units."},{"Start":"06:11.614 ","End":"06:15.410","Text":"Thus, when Theta is 0.5 radians,"},{"Start":"06:15.410 ","End":"06:19.100","Text":"the distance between the girls is changing at the rate"},{"Start":"06:19.100 ","End":"06:24.410","Text":"of approximately 0.498 feet per minute."},{"Start":"06:24.410 ","End":"06:26.555","Text":"There is something I prefer."},{"Start":"06:26.555 ","End":"06:28.670","Text":"If I know if it\u0027s increasing or decreasing,"},{"Start":"06:28.670 ","End":"06:29.870","Text":"I prefer to state that."},{"Start":"06:29.870 ","End":"06:31.520","Text":"So instead of changing,"},{"Start":"06:31.520 ","End":"06:36.455","Text":"what I would rather do is say increasing because it\u0027s positive."},{"Start":"06:36.455 ","End":"06:40.565","Text":"So it\u0027s increasing at the rate of so-and-so."},{"Start":"06:40.565 ","End":"06:44.280","Text":"That\u0027s it. We\u0027re done for this exercise."}],"ID":6296},{"Watched":false,"Name":"Exercise 4","Duration":"9m 53s","ChapterTopicVideoID":6285,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.205","Text":"Here we have a 2 part exercise in related rates."},{"Start":"00:05.205 ","End":"00:08.910","Text":"A tank of gasoline in the shape of a cone is"},{"Start":"00:08.910 ","End":"00:13.290","Text":"leaking at a constant rate of 3 cubic feet an hour."},{"Start":"00:13.290 ","End":"00:19.350","Text":"The base radius of the tank is 4 feet and the height of the tank is 13 feet."},{"Start":"00:19.350 ","End":"00:24.450","Text":"Part a, at what rate is the depth of the gasoline"},{"Start":"00:24.450 ","End":"00:29.520","Text":"in the tank changing when the depth of the gasoline is 5 foot?"},{"Start":"00:29.520 ","End":"00:34.410","Text":"b, at what rate is the radius of the top of the gasoline in"},{"Start":"00:34.410 ","End":"00:39.770","Text":"the tank changing when the depth of the gasoline is 5 feet?"},{"Start":"00:39.770 ","End":"00:43.940","Text":"I noticed first of all that it\u0027s an upside-down cone,"},{"Start":"00:43.940 ","End":"00:47.585","Text":"and actually what they call the base is what\u0027s at the top."},{"Start":"00:47.585 ","End":"00:51.590","Text":"Now what we\u0027re given is that this is 4,"},{"Start":"00:51.590 ","End":"01:00.795","Text":"and the units we\u0027re going to be talking in is feet for length and feet cubed for volume,"},{"Start":"01:00.795 ","End":"01:04.570","Text":"and hours when we talk about time."},{"Start":"01:04.570 ","End":"01:11.970","Text":"This is 4 feet radius of the base of the tank and the height of the tank is 13 feet."},{"Start":"01:11.970 ","End":"01:13.504","Text":"Now, both of these are not changing."},{"Start":"01:13.504 ","End":"01:16.400","Text":"We have the volume of the gasoline,"},{"Start":"01:16.400 ","End":"01:19.850","Text":"so that will be V. It is decreasing."},{"Start":"01:19.850 ","End":"01:25.565","Text":"There is also the radius of the base of the gasoline cone,"},{"Start":"01:25.565 ","End":"01:27.500","Text":"and this I\u0027ll call r,"},{"Start":"01:27.500 ","End":"01:31.160","Text":"and r is a function of time and it\u0027s also decreasing I know,"},{"Start":"01:31.160 ","End":"01:36.000","Text":"and this we\u0027ll call h and it\u0027s the height of the gasoline."},{"Start":"01:36.000 ","End":"01:41.490","Text":"V, r, and h we should write because they are variables which change over time,"},{"Start":"01:41.490 ","End":"01:43.020","Text":"and that\u0027s what I need here."},{"Start":"01:43.020 ","End":"01:49.130","Text":"V of t is the volume of gasoline as a function of time at"},{"Start":"01:49.130 ","End":"01:56.225","Text":"any time t. Then I have h of t which is the height or depth of the gasoline,"},{"Start":"01:56.225 ","End":"02:00.705","Text":"r of t is the radius of the gasoline."},{"Start":"02:00.705 ","End":"02:04.050","Text":"I\u0027ll write the units for each."},{"Start":"02:04.050 ","End":"02:07.865","Text":"The volume of the gasoline is in cubic feet."},{"Start":"02:07.865 ","End":"02:11.885","Text":"The depth and height are units of length and they\u0027re just in feet."},{"Start":"02:11.885 ","End":"02:15.940","Text":"Next I write a heading called hence because it follows from these,"},{"Start":"02:15.940 ","End":"02:19.180","Text":"it\u0027s just simply the derivatives of each of these."},{"Start":"02:19.180 ","End":"02:23.610","Text":"It becomes rate of change when you derive."},{"Start":"02:23.610 ","End":"02:32.240","Text":"V prime of t equals rate of change of volume of gasoline,"},{"Start":"02:32.240 ","End":"02:35.765","Text":"and I\u0027m writing everything in cubic feet per minute."},{"Start":"02:35.765 ","End":"02:38.480","Text":"In general, whenever you have these derivatives,"},{"Start":"02:38.480 ","End":"02:41.335","Text":"the derivative with respect to time,"},{"Start":"02:41.335 ","End":"02:44.235","Text":"the units are always per unit time."},{"Start":"02:44.235 ","End":"02:45.540","Text":"In this case per minute."},{"Start":"02:45.540 ","End":"02:47.990","Text":"H prime of t. Again,"},{"Start":"02:47.990 ","End":"02:52.995","Text":"it\u0027s the rate of change of the depth of the gasoline in feet per minute."},{"Start":"02:52.995 ","End":"02:55.640","Text":"Lastly, r prime of t,"},{"Start":"02:55.640 ","End":"03:00.275","Text":"rate of change of the radius of the gasoline in feet per minute."},{"Start":"03:00.275 ","End":"03:04.370","Text":"Next 2 headings will be what we\u0027re given and what we\u0027re looking for."},{"Start":"03:04.370 ","End":"03:09.500","Text":"What we\u0027re given in terms of quantities that are changing is simply that"},{"Start":"03:09.500 ","End":"03:14.805","Text":"the cone is leaking at a constant rate of 3 cubic feet an hour."},{"Start":"03:14.805 ","End":"03:19.640","Text":"In a we\u0027re looking for the rate of change of the depth of"},{"Start":"03:19.640 ","End":"03:24.485","Text":"the gasoline when the depth of the gasoline is 5 foot,"},{"Start":"03:24.485 ","End":"03:29.540","Text":"and in b we\u0027re looking for the rate of change of the radius of the top"},{"Start":"03:29.540 ","End":"03:35.045","Text":"at the moment when the depth of the gasoline is 15 foot."},{"Start":"03:35.045 ","End":"03:37.610","Text":"As for what we\u0027re looking for in a,"},{"Start":"03:37.610 ","End":"03:40.450","Text":"we\u0027re actually looking for h prime,"},{"Start":"03:40.450 ","End":"03:45.855","Text":"because the depth is h, and h prime when h is 5."},{"Start":"03:45.855 ","End":"03:50.565","Text":"Here we\u0027re looking for r prime also when h is 5."},{"Start":"03:50.565 ","End":"03:53.210","Text":"Let me write that down in mathematical terms now."},{"Start":"03:53.210 ","End":"03:54.635","Text":"Here in the given,"},{"Start":"03:54.635 ","End":"04:02.030","Text":"we have that V prime is minus 3 for all t and in"},{"Start":"04:02.030 ","End":"04:09.765","Text":"a we have that h prime is what when h equals 5."},{"Start":"04:09.765 ","End":"04:16.140","Text":"In b what we\u0027re looking for is r prime is what also when h equals 5."},{"Start":"04:16.140 ","End":"04:21.020","Text":"Next we\u0027ll be needing 1 or more equations to tie all these variables together."},{"Start":"04:21.020 ","End":"04:23.060","Text":"We have quite a lot of variables,"},{"Start":"04:23.060 ","End":"04:25.740","Text":"so I know that this time we\u0027ll need 2 equations,"},{"Start":"04:25.740 ","End":"04:27.200","Text":"1 just won\u0027t be enough."},{"Start":"04:27.200 ","End":"04:31.400","Text":"The most obvious equation we have is the 1 from solid geometry that"},{"Start":"04:31.400 ","End":"04:35.735","Text":"relates the volume of a cone to its radius."},{"Start":"04:35.735 ","End":"04:43.440","Text":"We have that V equals 1/3Pi r squared h. Here we\u0027re going to be given h,"},{"Start":"04:43.440 ","End":"04:48.260","Text":"because we say when h is 5 but we won\u0027t know r and it might be"},{"Start":"04:48.260 ","End":"04:53.740","Text":"somewhere computing it unless we get another equation because this is really not enough."},{"Start":"04:53.740 ","End":"04:58.400","Text":"But the other equation that we can note if we go back to"},{"Start":"04:58.400 ","End":"05:03.440","Text":"the diagram is that we have actually some similar triangles."},{"Start":"05:03.440 ","End":"05:08.060","Text":"This whole triangle is the same as the highlighted triangle,"},{"Start":"05:08.060 ","End":"05:10.010","Text":"but we can take half a triangle."},{"Start":"05:10.010 ","End":"05:16.695","Text":"In other words, this part which I\u0027m contouring here will be similar to this 1 here."},{"Start":"05:16.695 ","End":"05:22.280","Text":"With similar triangles, the ratios of sides corresponding is the same."},{"Start":"05:22.280 ","End":"05:29.585","Text":"What we get, we can get r over h is going to be the same as 4 over 13."},{"Start":"05:29.585 ","End":"05:38.895","Text":"We can write that r over h is equal to 4 over 13,"},{"Start":"05:38.895 ","End":"05:44.325","Text":"and then usually I proceed to the derive heading and then the solve."},{"Start":"05:44.325 ","End":"05:45.870","Text":"See I don\u0027t really want the r here,"},{"Start":"05:45.870 ","End":"05:48.875","Text":"I want the connection between V and h. If I replace,"},{"Start":"05:48.875 ","End":"05:55.035","Text":"this gives me that r is equal to 4h over 13,"},{"Start":"05:55.035 ","End":"05:57.920","Text":"and then if I put this in here for r,"},{"Start":"05:57.920 ","End":"06:04.730","Text":"then I\u0027ll get that this implies that V is equal to 1/3Pi."},{"Start":"06:04.730 ","End":"06:10.710","Text":"Instead of r I\u0027ll put 4h over 13 squared times h,"},{"Start":"06:10.710 ","End":"06:13.395","Text":"and this is much better because I don\u0027t have r I have"},{"Start":"06:13.395 ","End":"06:17.445","Text":"a direct tie in between V and h. Let\u0027s take the constants together."},{"Start":"06:17.445 ","End":"06:19.695","Text":"We have 1/3 of Pi,"},{"Start":"06:19.695 ","End":"06:22.950","Text":"and then 4 squared is 16,"},{"Start":"06:22.950 ","End":"06:26.775","Text":"and 13 squared is 169,"},{"Start":"06:26.775 ","End":"06:30.675","Text":"169 times 3 is 507,"},{"Start":"06:30.675 ","End":"06:32.685","Text":"and there\u0027s still a Pi up here,"},{"Start":"06:32.685 ","End":"06:38.175","Text":"and then we just have h squared times h which is h cubed."},{"Start":"06:38.175 ","End":"06:41.180","Text":"It\u0027s just quite simply a constant times h cubed."},{"Start":"06:41.180 ","End":"06:44.330","Text":"Now I can go ahead and do the derive part."},{"Start":"06:44.330 ","End":"06:49.400","Text":"That\u0027ll give me that V is equal to"},{"Start":"06:49.400 ","End":"06:55.350","Text":"16Pi over 169 times h squared,"},{"Start":"06:55.350 ","End":"06:58.015","Text":"because the h cubed is 3h squared,"},{"Start":"06:58.015 ","End":"07:00.785","Text":"but we still have to multiply by h prime."},{"Start":"07:00.785 ","End":"07:02.435","Text":"I wrote V instead of V prime."},{"Start":"07:02.435 ","End":"07:06.290","Text":"We do know h because we\u0027re taking it when h equals 5,"},{"Start":"07:06.290 ","End":"07:09.190","Text":"and that will give us the h prime that we want."},{"Start":"07:09.190 ","End":"07:13.695","Text":"Then after h prime we\u0027ll then go back and also get r prime."},{"Start":"07:13.695 ","End":"07:18.050","Text":"Now onto the substitute part before this line disappears,"},{"Start":"07:18.050 ","End":"07:21.125","Text":"just notice V prime is minus 3."},{"Start":"07:21.125 ","End":"07:28.800","Text":"I can already say that minus 3 equals 16Pi over 169,"},{"Start":"07:28.800 ","End":"07:33.530","Text":"h we know is 5 because that\u0027s what we\u0027re looking for is the rate of"},{"Start":"07:33.530 ","End":"07:39.960","Text":"change of h prime when h is 5 squared times h prime."},{"Start":"07:39.960 ","End":"07:45.235","Text":"We only have 1 variable to solve here and that\u0027s h prime."},{"Start":"07:45.235 ","End":"07:49.505","Text":"H prime is, just put all the stuff onto the other side."},{"Start":"07:49.505 ","End":"07:56.590","Text":"We see we have a minus 3 and then we have 169 over 16Pi."},{"Start":"07:56.590 ","End":"08:01.790","Text":"This 5 squared also goes into the denominator there which is 25."},{"Start":"08:01.790 ","End":"08:05.945","Text":"I\u0027m glad to know that we have a negative number here because as I said,"},{"Start":"08:05.945 ","End":"08:07.625","Text":"V prime is negative,"},{"Start":"08:07.625 ","End":"08:08.915","Text":"and all the others,"},{"Start":"08:08.915 ","End":"08:10.100","Text":"the volume, the height,"},{"Start":"08:10.100 ","End":"08:12.320","Text":"and the radius are all shrinking."},{"Start":"08:12.320 ","End":"08:18.525","Text":"All that remains is to compute this using a calculator and this was part a."},{"Start":"08:18.525 ","End":"08:21.479","Text":"In part b, we have something similar."},{"Start":"08:21.479 ","End":"08:25.460","Text":"We want to know what is r prime at the same moment."},{"Start":"08:25.460 ","End":"08:29.765","Text":"Well, I\u0027m going to use an observation and I already highlighted it here that"},{"Start":"08:29.765 ","End":"08:34.290","Text":"r is given in terms of h as 4/13 of h,"},{"Start":"08:34.290 ","End":"08:37.475","Text":"so if I just differentiate this with respect to time,"},{"Start":"08:37.475 ","End":"08:43.785","Text":"then r prime equal to 4/13 of h prime."},{"Start":"08:43.785 ","End":"08:48.770","Text":"I can either just take this and do it all on the calculator or I can take"},{"Start":"08:48.770 ","End":"08:53.870","Text":"this and put another 4/13 minus 3, 169 times."},{"Start":"08:53.870 ","End":"08:55.190","Text":"Here there\u0027ll be an extra 4,"},{"Start":"08:55.190 ","End":"08:57.930","Text":"here there\u0027ll be an extra 13,"},{"Start":"08:58.310 ","End":"09:03.365","Text":"16Pi times 25, and things do cancel,"},{"Start":"09:03.365 ","End":"09:07.265","Text":"minus 39 over 100Pi."},{"Start":"09:07.265 ","End":"09:09.650","Text":"If I do this on the calculator,"},{"Start":"09:09.650 ","End":"09:14.930","Text":"I\u0027ll get minus 0.39 over Pi and that\u0027s easy enough to punch"},{"Start":"09:14.930 ","End":"09:23.155","Text":"in and what I get is minus 0.124,"},{"Start":"09:23.155 ","End":"09:27.765","Text":"and that\u0027ll also be in feet per hour."},{"Start":"09:27.765 ","End":"09:30.740","Text":"These are the 2 answers we were looking for."},{"Start":"09:30.740 ","End":"09:34.930","Text":"I\u0027ll skip the stage of writing it in nice words and units"},{"Start":"09:34.930 ","End":"09:39.515","Text":"and I would say when the depth of the gasoline is 6 feet,"},{"Start":"09:39.515 ","End":"09:44.120","Text":"the rate of change of the depth is so and"},{"Start":"09:44.120 ","End":"09:48.920","Text":"so feet per hour and the rate of change of the radius is so and so feet per hour,"},{"Start":"09:48.920 ","End":"09:51.500","Text":"that would be nice just skip that this time."},{"Start":"09:51.500 ","End":"09:54.029","Text":"We\u0027re done with this 1."}],"ID":6297},{"Watched":false,"Name":"Exercise 5","Duration":"8m 21s","ChapterTopicVideoID":6286,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"This is an exercise in related rates."},{"Start":"00:03.120 ","End":"00:06.030","Text":"A trough of water is 9 meters deep and"},{"Start":"00:06.030 ","End":"00:09.630","Text":"its ends are in the shape of an isosceles triangle,"},{"Start":"00:09.630 ","End":"00:14.295","Text":"whose width is 6 meters and the height is 3 meters."},{"Start":"00:14.295 ","End":"00:20.160","Text":"If water is being pumped in at a constant rate of 6 cubic meters a second,"},{"Start":"00:20.160 ","End":"00:23.895","Text":"at what rate is the height of the water changing,"},{"Start":"00:23.895 ","End":"00:27.870","Text":"when the water has a height of a 110 centimeters?"},{"Start":"00:27.870 ","End":"00:29.910","Text":"They\u0027ve given us a diagram,"},{"Start":"00:29.910 ","End":"00:34.080","Text":"we just need to fill in some of the numbers and variable names."},{"Start":"00:34.080 ","End":"00:38.670","Text":"What I want to do is my first section is usually what I call denote,"},{"Start":"00:38.670 ","End":"00:42.920","Text":"that\u0027s where I name which variables I\u0027m using and what they mean."},{"Start":"00:42.920 ","End":"00:47.600","Text":"Under the denote here I\u0027ll be writing some variable names,"},{"Start":"00:47.600 ","End":"00:50.900","Text":"especially those variables which are changing with time."},{"Start":"00:50.900 ","End":"00:53.315","Text":"Let\u0027s first of all mark some things in the sketch."},{"Start":"00:53.315 ","End":"00:56.855","Text":"As I recall, this was 9."},{"Start":"00:56.855 ","End":"01:01.235","Text":"I\u0027ll mention that the units of length I\u0027m going to take is going to be meters."},{"Start":"01:01.235 ","End":"01:03.920","Text":"There\u0027s just 1 place we have to watch out where they\u0027ve used"},{"Start":"01:03.920 ","End":"01:06.690","Text":"centimeters so we have to convert that to meters,"},{"Start":"01:06.690 ","End":"01:13.010","Text":"but I\u0027ll take length in meters and time in seconds and of course volume in cubic meters."},{"Start":"01:13.010 ","End":"01:19.700","Text":"We\u0027ve got this as 6 and we\u0027ve got this as 3 and these are the dimensions of the trough."},{"Start":"01:19.700 ","End":"01:24.780","Text":"Now, the water is changing and the water has a height,"},{"Start":"01:24.780 ","End":"01:27.450","Text":"so how about h for height?"},{"Start":"01:27.450 ","End":"01:31.380","Text":"It has a width, so how about w for width?"},{"Start":"01:31.380 ","End":"01:36.650","Text":"All together, the water has a volume so volume will be in meters cube,"},{"Start":"01:36.650 ","End":"01:39.770","Text":"so that will be V and each of these is changing over time."},{"Start":"01:39.770 ","End":"01:45.560","Text":"What I\u0027ll write is that V equals volume of the water in"},{"Start":"01:45.560 ","End":"01:49.310","Text":"the trough and the units that\u0027s going to be in"},{"Start":"01:49.310 ","End":"01:53.375","Text":"cubic meters and we\u0027re going to have h for the height,"},{"Start":"01:53.375 ","End":"01:55.865","Text":"which is the height of the water,"},{"Start":"01:55.865 ","End":"01:57.905","Text":"this will be in meters."},{"Start":"01:57.905 ","End":"01:59.660","Text":"Similarly, we\u0027ll have w,"},{"Start":"01:59.660 ","End":"02:02.840","Text":"which is the width of water and that\u0027s also in"},{"Start":"02:02.840 ","End":"02:07.580","Text":"meters and then I usually have a section called hence,"},{"Start":"02:07.580 ","End":"02:11.090","Text":"cause I derive it from these V prime."},{"Start":"02:11.090 ","End":"02:12.920","Text":"These are all functions of t,"},{"Start":"02:12.920 ","End":"02:15.740","Text":"but like I said earlier we\u0027re going to skip"},{"Start":"02:15.740 ","End":"02:21.410","Text":"the parentheses t just to clean things up and we know that these are functions of time."},{"Start":"02:21.410 ","End":"02:28.010","Text":"V prime is the rate of change of volume and by volume I mean"},{"Start":"02:28.010 ","End":"02:35.240","Text":"the volume of the water in the trough and then I have h prime and w prime,"},{"Start":"02:35.240 ","End":"02:36.470","Text":"1 is the rate of change of height,"},{"Start":"02:36.470 ","End":"02:39.095","Text":"1 is the rate of change of width, I\u0027ll just write it in."},{"Start":"02:39.095 ","End":"02:40.430","Text":"There we are."},{"Start":"02:40.430 ","End":"02:46.640","Text":"All these derivatives are with respect to t. The rate of change, this of course,"},{"Start":"02:46.640 ","End":"02:50.390","Text":"is the same as these except divided by the unit of time so this"},{"Start":"02:50.390 ","End":"02:54.260","Text":"will be in cubic meters a second,"},{"Start":"02:54.260 ","End":"02:58.715","Text":"rate of change of height will be in meters per"},{"Start":"02:58.715 ","End":"03:05.090","Text":"second and the same with the width in meters per second."},{"Start":"03:05.090 ","End":"03:08.120","Text":"The next section I go for is"},{"Start":"03:08.120 ","End":"03:11.350","Text":"what we\u0027re given and then after that, what we\u0027re looking for."},{"Start":"03:11.350 ","End":"03:13.820","Text":"I\u0027ve written both of these headings given and looking for."},{"Start":"03:13.820 ","End":"03:17.390","Text":"What we\u0027re given is that V prime,"},{"Start":"03:17.390 ","End":"03:19.340","Text":"which is the rate of change of the volume."},{"Start":"03:19.340 ","End":"03:22.205","Text":"It\u0027s the rate at which the volume is being pumped in,"},{"Start":"03:22.205 ","End":"03:29.480","Text":"so V prime equals 6 and that\u0027s a constant doesn\u0027t depend on t and it\u0027s being pumped in."},{"Start":"03:29.480 ","End":"03:32.840","Text":"Yes, positive and we\u0027re looking for h prime,"},{"Start":"03:32.840 ","End":"03:37.625","Text":"we\u0027re looking for the rate of change of h. When I remember it said"},{"Start":"03:37.625 ","End":"03:43.705","Text":"a 110 centimeter when h is 1.1 meters."},{"Start":"03:43.705 ","End":"03:49.265","Text":"The next section that we want is an equation which connects all these variables."},{"Start":"03:49.265 ","End":"03:54.095","Text":"The main equation will be the volume of the trough,"},{"Start":"03:54.095 ","End":"03:56.840","Text":"which is actually a triangular prism."},{"Start":"03:56.840 ","End":"04:01.460","Text":"It\u0027s the area of the triangle times the depth or length."},{"Start":"04:01.460 ","End":"04:08.120","Text":"The volume of this prism will be the area and we\u0027re talking about the changing volume,"},{"Start":"04:08.120 ","End":"04:12.380","Text":"so it\u0027s the blue, the water is the area of this triangle."},{"Start":"04:12.380 ","End":"04:13.715","Text":"Yeah, I\u0027ll even shade it,"},{"Start":"04:13.715 ","End":"04:17.660","Text":"times the length of it which is what they call depth,"},{"Start":"04:17.660 ","End":"04:19.870","Text":"which will be the 9."},{"Start":"04:19.870 ","End":"04:27.050","Text":"Let\u0027s write that down and we will get that V is equal to,"},{"Start":"04:27.050 ","End":"04:32.225","Text":"select the area of the base times height for any prism now this area of a triangle."},{"Start":"04:32.225 ","End":"04:34.955","Text":"This bit here, this was w,"},{"Start":"04:34.955 ","End":"04:42.660","Text":"so area of a triangle is half base times height so we have 1/2 w times h,"},{"Start":"04:42.660 ","End":"04:49.055","Text":"but then times 8 which is the unchanging length and they call it depth."},{"Start":"04:49.055 ","End":"04:51.125","Text":"That\u0027s our main equation,"},{"Start":"04:51.125 ","End":"04:54.650","Text":"but there is another equation I\u0027d like to write."},{"Start":"04:54.650 ","End":"04:58.160","Text":"I just gained experience from the previous question which also"},{"Start":"04:58.160 ","End":"05:02.060","Text":"had similar triangles and I\u0027d like to write that as an equation."},{"Start":"05:02.060 ","End":"05:05.990","Text":"What I want to say is that we have this triangle that I\u0027ve shaded"},{"Start":"05:05.990 ","End":"05:10.205","Text":"is similar to this triangle which is the side of the trough."},{"Start":"05:10.205 ","End":"05:17.510","Text":"What I can say is that if I take this width and I divide it by the whole width,"},{"Start":"05:17.510 ","End":"05:24.125","Text":"which is the 6 but this is the same by similar triangles to this h,"},{"Start":"05:24.125 ","End":"05:27.560","Text":"h in of the water over 3 of the trough for"},{"Start":"05:27.560 ","End":"05:31.310","Text":"the numerators of relate to the water and the denominators are from the trough."},{"Start":"05:31.310 ","End":"05:32.720","Text":"I would like to write this as"},{"Start":"05:32.720 ","End":"05:36.350","Text":"another equation because otherwise they\u0027ll have too many equations here and this will"},{"Start":"05:36.350 ","End":"05:38.870","Text":"enable me to write w in terms of h or the other way"},{"Start":"05:38.870 ","End":"05:41.870","Text":"around and last thing where your indulgence,"},{"Start":"05:41.870 ","End":"05:44.550","Text":"I didn\u0027t want to start developing this later."},{"Start":"05:44.550 ","End":"05:49.340","Text":"I can multiply 8 by a 1/2 and let\u0027s write this as already as 4 and"},{"Start":"05:49.340 ","End":"05:54.305","Text":"now we come to the section on derive where we will get an extra information."},{"Start":"05:54.305 ","End":"05:56.015","Text":"But I look at this and I say,"},{"Start":"05:56.015 ","End":"05:57.350","Text":"okay, I want to derive this,"},{"Start":"05:57.350 ","End":"05:59.450","Text":"but I really wanted V in terms of"},{"Start":"05:59.450 ","End":"06:04.005","Text":"1 variable and preferably h because that\u0027s what I\u0027m looking for."},{"Start":"06:04.005 ","End":"06:06.620","Text":"Now, remember now that, that\u0027s why I put in"},{"Start":"06:06.620 ","End":"06:09.875","Text":"this extra equation from the similar triangles."},{"Start":"06:09.875 ","End":"06:12.890","Text":"Before I derive, just indulge me here,"},{"Start":"06:12.890 ","End":"06:15.015","Text":"I\u0027ll do a little bit of algebra."},{"Start":"06:15.015 ","End":"06:20.840","Text":"From this, if I multiply both sides by 6 and remembering that 6 over 3 is 2,"},{"Start":"06:20.840 ","End":"06:25.755","Text":"I get that w is equal to 2h."},{"Start":"06:25.755 ","End":"06:32.105","Text":"The second thing I could tell now is that if I put w equals 2h here,"},{"Start":"06:32.105 ","End":"06:35.895","Text":"I get that V is 2 times 4 is 8."},{"Start":"06:35.895 ","End":"06:39.865","Text":"I get that V is equal to 8,"},{"Start":"06:39.865 ","End":"06:44.550","Text":"hh is h squared and this is very useful to me."},{"Start":"06:44.550 ","End":"06:48.560","Text":"Now, I\u0027ll do the deriving and I\u0027ll derive both of them."},{"Start":"06:48.560 ","End":"06:50.300","Text":"I don\u0027t know if I\u0027ll need the second 1,"},{"Start":"06:50.300 ","End":"06:55.220","Text":"but the first 1 I\u0027ll get will be that V prime is equal"},{"Start":"06:55.220 ","End":"07:03.695","Text":"to 16h times h prime because it\u0027s also a function of t and the other 1,"},{"Start":"07:03.695 ","End":"07:06.500","Text":"each w prime is 2h prime."},{"Start":"07:06.500 ","End":"07:10.850","Text":"Next section is to substitute variables that we know."},{"Start":"07:10.850 ","End":"07:13.435","Text":"Let\u0027s go for substitute now."},{"Start":"07:13.435 ","End":"07:15.470","Text":"We know everything except h prime,"},{"Start":"07:15.470 ","End":"07:16.730","Text":"which is what we\u0027re looking for."},{"Start":"07:16.730 ","End":"07:19.565","Text":"We\u0027re looking for it when h is 1.1,"},{"Start":"07:19.565 ","End":"07:22.115","Text":"so V prime is a constant,"},{"Start":"07:22.115 ","End":"07:25.005","Text":"it\u0027s positive, it\u0027s 6 always,"},{"Start":"07:25.005 ","End":"07:29.090","Text":"that\u0027s the rate at which the water is being pumped in is equal to 16 is"},{"Start":"07:29.090 ","End":"07:35.445","Text":"just a number h is the 1.1 and times h prime."},{"Start":"07:35.445 ","End":"07:40.555","Text":"Now, we just have to take h prime out and we get"},{"Start":"07:40.555 ","End":"07:49.110","Text":"h prime is equal to 6 over 16 times 1.1,"},{"Start":"07:49.110 ","End":"07:56.055","Text":"is 17.6 and this is equal to I\u0027m just going to look in my calculator."},{"Start":"07:56.055 ","End":"08:02.960","Text":"Approximately 0.341 to 3 decimal places I took it and this is in meters per second."},{"Start":"08:02.960 ","End":"08:07.715","Text":"Finally, I just write something in words, something human-readable."},{"Start":"08:07.715 ","End":"08:12.320","Text":"The height of the water is changing at a rate of approximately"},{"Start":"08:12.320 ","End":"08:19.770","Text":"0.341 meters a second and that\u0027s more human-readable, I\u0027m just leaving it like that."},{"Start":"08:19.770 ","End":"08:22.600","Text":"We\u0027re done, we\u0027ve answered the question."}],"ID":6298},{"Watched":false,"Name":"Exercise 6","Duration":"15m 33s","ChapterTopicVideoID":1454,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"Here we have an exercise in related rates, and we should read it first."},{"Start":"00:05.370 ","End":"00:09.540","Text":"A light is on the top of a 13-foot tall pole and"},{"Start":"00:09.540 ","End":"00:15.645","Text":"a 5-foot tall girl is running away from the pole at a rate of 3 feet a second."},{"Start":"00:15.645 ","End":"00:19.500","Text":"Part a, at what rate is the tip of the shadow"},{"Start":"00:19.500 ","End":"00:24.135","Text":"moving away from the pole, when the girl is 30-foot from the pole?"},{"Start":"00:24.135 ","End":"00:28.380","Text":"Part b, at what rate is the tip of the shadow moving away from"},{"Start":"00:28.380 ","End":"00:34.105","Text":"the girl, when the girl is 30-foot from the pole?"},{"Start":"00:34.105 ","End":"00:36.275","Text":"Very similar, a and b,"},{"Start":"00:36.275 ","End":"00:38.180","Text":"they both occur at the same time."},{"Start":"00:38.180 ","End":"00:39.350","Text":"Just in 1 case,"},{"Start":"00:39.350 ","End":"00:45.360","Text":"we want shadow away from the pole and in the other case shadow moving away from the girl."},{"Start":"00:45.770 ","End":"00:50.170","Text":"Let\u0027s get onto the solution."},{"Start":"00:51.800 ","End":"00:55.850","Text":"I\u0027m going to combine 2 headings into 1,"},{"Start":"00:55.850 ","End":"00:59.255","Text":"the sketch part because I noticed there\u0027s a sketch down there."},{"Start":"00:59.255 ","End":"01:01.710","Text":"Let\u0027s see if we can get to it."},{"Start":"01:04.400 ","End":"01:08.315","Text":"Denote some quantity by a variable name,"},{"Start":"01:08.315 ","End":"01:12.930","Text":"together with the graph, the sketch."},{"Start":"01:12.980 ","End":"01:16.640","Text":"I premarked some of them."},{"Start":"01:16.640 ","End":"01:22.085","Text":"I took X to denote the distance from the lamp"},{"Start":"01:22.085 ","End":"01:27.660","Text":"to the tip of the shadow, and then I took an Xg,"},{"Start":"01:27.660 ","End":"01:29.415","Text":"this is a g for girl,"},{"Start":"01:29.415 ","End":"01:33.469","Text":"to be the distance from the lamp to the girl, and Xs,"},{"Start":"01:33.469 ","End":"01:38.210","Text":"that\u0027s for shadow, be the distance from the girl to the tip of the shadow,"},{"Start":"01:38.210 ","End":"01:41.000","Text":"in other words, the length of the shadow."},{"Start":"01:41.000 ","End":"01:45.335","Text":"There\u0027s also a couple of constants."},{"Start":"01:45.335 ","End":"01:51.805","Text":"We have the height of the lamp as 13."},{"Start":"01:51.805 ","End":"01:57.850","Text":"Yes, the height of the girl is 5 feet."},{"Start":"01:59.600 ","End":"02:03.560","Text":"I think that\u0027s all we\u0027ll need except that there\u0027s"},{"Start":"02:03.560 ","End":"02:07.220","Text":"1 other thing that\u0027s given to us, and that is that"},{"Start":"02:07.220 ","End":"02:15.020","Text":"the girl is moving away from the pole in this direction."},{"Start":"02:15.020 ","End":"02:19.355","Text":"We\u0027re given that the rate she\u0027s moving at is"},{"Start":"02:19.355 ","End":"02:24.160","Text":"3 feet per second."},{"Start":"02:24.160 ","End":"02:27.460","Text":"In general, you can put the units everywhere,"},{"Start":"02:27.460 ","End":"02:34.675","Text":"but we see from this that we are going to be measuring distance in feet and length."},{"Start":"02:34.675 ","End":"02:38.255","Text":"Sorry, that is length."},{"Start":"02:38.255 ","End":"02:43.505","Text":"Units of length are in feet and units of time are in seconds."},{"Start":"02:43.505 ","End":"02:48.030","Text":"There\u0027s nothing else but distance and time."},{"Start":"02:49.610 ","End":"02:57.890","Text":"First thing I usually do with this is denote variables,"},{"Start":"02:57.890 ","End":"03:00.065","Text":"the ones that change with time."},{"Start":"03:00.065 ","End":"03:01.895","Text":"I have 3 of them here."},{"Start":"03:01.895 ","End":"03:04.895","Text":"I have X, I\u0027ll just write their names first,"},{"Start":"03:04.895 ","End":"03:08.329","Text":"I have X for the girl,"},{"Start":"03:08.329 ","End":"03:10.550","Text":"and I have X for the shadow."},{"Start":"03:10.550 ","End":"03:14.855","Text":"All these 3 are units of length and they\u0027re going to be in feet."},{"Start":"03:14.855 ","End":"03:23.710","Text":"But X is going to be the distance from the lamp to the tip of the shadow."},{"Start":"03:23.710 ","End":"03:26.270","Text":"I\u0027m just going to write all these."},{"Start":"03:26.370 ","End":"03:29.124","Text":"Here I\u0027ve just written them out."},{"Start":"03:29.124 ","End":"03:33.020","Text":"X is the distance from the lamp to the tip of the shadow,"},{"Start":"03:33.020 ","End":"03:35.700","Text":"Xg, from the lamp to the girl,"},{"Start":"03:35.700 ","End":"03:38.970","Text":"Xs, length of the shadow, and so on."},{"Start":"03:38.970 ","End":"03:44.590","Text":"Usually, under the Hence section,"},{"Start":"03:44.590 ","End":"03:47.320","Text":"I usually put the derivatives of these,"},{"Start":"03:47.320 ","End":"03:52.940","Text":"but I don\u0027t think that spelling out what it means is going to give me any insight,"},{"Start":"03:52.940 ","End":"04:00.595","Text":"so I\u0027ll just write like this quickly the derivatives of all these,"},{"Start":"04:00.595 ","End":"04:01.945","Text":"each 1 of them,"},{"Start":"04:01.945 ","End":"04:03.250","Text":"I\u0027ll put them all together"},{"Start":"04:03.250 ","End":"04:06.620","Text":"as the rate of change"},{"Start":"04:10.610 ","End":"04:14.100","Text":"of 3 above of"},{"Start":"04:14.100 ","End":"04:17.540","Text":"the X prime is the rate of change of"},{"Start":"04:17.540 ","End":"04:19.190","Text":"X. I\u0027m not going to say it\u0027s rate of change of"},{"Start":"04:19.190 ","End":"04:21.040","Text":"the distance from the lamp to the tip of the shadow,"},{"Start":"04:21.040 ","End":"04:24.150","Text":"and so on, we\u0027ll just say that these are the derivatives,"},{"Start":"04:24.150 ","End":"04:29.690","Text":"and realize that derivative just means rate of change,"},{"Start":"04:29.690 ","End":"04:33.350","Text":"or the rate of change of a distance is often called a speed."},{"Start":"04:33.350 ","End":"04:37.700","Text":"We\u0027ll just leave it at that for this heading,"},{"Start":"04:37.700 ","End":"04:41.610","Text":"which I put in usually a section."},{"Start":"04:41.710 ","End":"04:47.330","Text":"The next couple of headings that I usually like to put in are what were"},{"Start":"04:47.330 ","End":"04:53.390","Text":"given and what we\u0027re looking for, very important headings."},{"Start":"04:56.150 ","End":"04:59.310","Text":"What were given is we\u0027re given,"},{"Start":"04:59.310 ","End":"05:05.855","Text":"if you remember, that Xg prime"},{"Start":"05:05.855 ","End":"05:12.140","Text":"or perhaps I should have pointed out in the picture that this 3 feet a second is"},{"Start":"05:12.140 ","End":"05:18.630","Text":"actually the rate at which this Xg is changing,"},{"Start":"05:18.630 ","End":"05:22.395","Text":"which is the distance of the girl from the lamp."},{"Start":"05:22.395 ","End":"05:28.575","Text":"This is Xg prime. There we are."},{"Start":"05:28.575 ","End":"05:33.695","Text":"That is equal to a constant 3 feet per second."},{"Start":"05:33.695 ","End":"05:35.960","Text":"We\u0027re looking for actually 2 things,"},{"Start":"05:35.960 ","End":"05:42.655","Text":"we\u0027ll call them a and b. a is X prime,"},{"Start":"05:42.655 ","End":"05:45.860","Text":"because it relates to the distance of the tip of"},{"Start":"05:45.860 ","End":"05:49.235","Text":"the shadow from the lamp and how fast it\u0027s moving."},{"Start":"05:49.235 ","End":"05:56.880","Text":"When equals 30 means the girl is 30-foot away from the pole."},{"Start":"05:58.670 ","End":"06:06.795","Text":"In part b, what we want is Xs prime,"},{"Start":"06:06.795 ","End":"06:09.990","Text":"because Xs is the length of the shadow, and"},{"Start":"06:09.990 ","End":"06:14.340","Text":"we\u0027re asking how fast the length of the shadow is changing."},{"Start":"06:14.710 ","End":"06:17.840","Text":"Again, at the same time,"},{"Start":"06:17.840 ","End":"06:25.390","Text":"which is when the girl is 30 feet away from the wall."},{"Start":"06:26.960 ","End":"06:30.920","Text":"Next section, next heading is"},{"Start":"06:30.920 ","End":"06:38.590","Text":"equation to connect some of the variables together. Let\u0027s see."},{"Start":"06:39.320 ","End":"06:43.960","Text":"Call these equations, and there is actually more than 1."},{"Start":"06:43.960 ","End":"06:45.770","Text":"1 of them is a very obvious 1."},{"Start":"06:45.770 ","End":"06:47.195","Text":"Let me just write it."},{"Start":"06:47.195 ","End":"06:53.970","Text":"X equals Xg plus Xs,"},{"Start":"06:53.970 ","End":"06:56.780","Text":"which is trivial to look at."},{"Start":"06:56.780 ","End":"06:58.010","Text":"If you look at the graph,"},{"Start":"06:58.010 ","End":"07:02.950","Text":"just says that this total X is made up of these 2 sub X\u0027s."},{"Start":"07:02.950 ","End":"07:06.440","Text":"The second, we\u0027ll come back to this picture,"},{"Start":"07:06.440 ","End":"07:09.770","Text":"will be related to similar triangles."},{"Start":"07:09.770 ","End":"07:15.300","Text":"First, this 1, that X is Xg plus Xs."},{"Start":"07:16.500 ","End":"07:24.580","Text":"We do have similar triangles."},{"Start":"07:24.580 ","End":"07:25.780","Text":"We have the large 1."},{"Start":"07:25.780 ","End":"07:28.990","Text":"You know what, I\u0027ll just color them, and you see"},{"Start":"07:28.990 ","End":"07:33.940","Text":"this red triangle is similar to the blue triangle."},{"Start":"07:33.940 ","End":"07:39.415","Text":"What I\u0027m going to get is if I take the blue over the red,"},{"Start":"07:39.415 ","End":"07:47.890","Text":"then 5 over 13 is going to equal Xs over X."},{"Start":"07:47.890 ","End":"07:50.305","Text":"Let\u0027s write that down there."},{"Start":"07:50.305 ","End":"07:57.160","Text":"5 over 13 is equal to"},{"Start":"07:57.160 ","End":"08:04.570","Text":"Xs over X, and"},{"Start":"08:04.570 ","End":"08:11.425","Text":"we could also write that as Xs since X is made up of the sum of the 2,"},{"Start":"08:11.425 ","End":"08:18.520","Text":"we can get Xs plus Xg."},{"Start":"08:18.520 ","End":"08:23.710","Text":"These are our equations, and the next thing we want to do with the equation"},{"Start":"08:23.710 ","End":"08:29.760","Text":"is to get a bit of extra equation by deriving these."},{"Start":"08:29.760 ","End":"08:31.260","Text":"That\u0027s often, not always,"},{"Start":"08:31.260 ","End":"08:33.165","Text":"but usually that\u0027s what you want to do."},{"Start":"08:33.165 ","End":"08:37.700","Text":"I\u0027m going to call this section"},{"Start":"08:37.700 ","End":"08:44.380","Text":"Derive, and I\u0027m going to say here that the derivative of this 1,"},{"Start":"08:44.380 ","End":"08:47.710","Text":"all though it\u0027s trivial to our main equation, is"},{"Start":"08:47.710 ","End":"08:53.155","Text":"that X prime and all the derivatives with respect to t"},{"Start":"08:53.155 ","End":"09:02.890","Text":"is equal to Xg prime plus Xs prime and the other equation,"},{"Start":"09:02.890 ","End":"09:08.605","Text":"well, I\u0027ll leave that for now as far as deriving in case we need it,"},{"Start":"09:08.605 ","End":"09:10.345","Text":"but we may or may not."},{"Start":"09:10.345 ","End":"09:14.740","Text":"In any event, what I really intend to do is to eliminate 1 of"},{"Start":"09:14.740 ","End":"09:21.880","Text":"these variables and just write X in terms of just Xg."},{"Start":"09:21.880 ","End":"09:24.429","Text":"In fact, I\u0027m going to make up a new heading"},{"Start":"09:24.429 ","End":"09:26.800","Text":"because I want to eliminate 1 of the variables."},{"Start":"09:26.800 ","End":"09:30.085","Text":"Let\u0027s just call it Eliminate."},{"Start":"09:30.085 ","End":"09:34.600","Text":"It doesn\u0027t always crop up,"},{"Start":"09:34.600 ","End":"09:40.150","Text":"but I\u0027ve noticed it\u0027s happened more than once that I want to eliminate a variable."},{"Start":"09:40.150 ","End":"09:45.670","Text":"In this case, I want to have just X and Xg together."},{"Start":"09:45.670 ","End":"09:52.975","Text":"What I\u0027ll do is I\u0027ll just take the first and the last and cross-multiply at first."},{"Start":"09:52.975 ","End":"10:04.060","Text":"What I\u0027ll get is 5 times Xs plus"},{"Start":"10:04.060 ","End":"10:09.440","Text":"Xg is equal to"},{"Start":"10:10.710 ","End":"10:17.380","Text":"13Xs, and with a little bit of algebra,"},{"Start":"10:17.380 ","End":"10:21.100","Text":"this will give me that Xs"},{"Start":"10:21.100 ","End":"10:29.330","Text":"is 5/8 of Xg."},{"Start":"10:29.760 ","End":"10:34.390","Text":"If I now plug this into this equation,"},{"Start":"10:34.390 ","End":"10:44.455","Text":"I get that X equals Xg plus 5/8 Xg and mentally I\u0027m doing 1 plus 5/8 and getting 13/8."},{"Start":"10:44.455 ","End":"10:51.295","Text":"In other words, this gives me that X equals 13"},{"Start":"10:51.295 ","End":"11:01.750","Text":"over 8 Xg, and actually this is the 1 that I would like to derive here and not this."},{"Start":"11:01.750 ","End":"11:09.009","Text":"This is what I\u0027m going to derive and get that X prime is equal"},{"Start":"11:09.009 ","End":"11:16.555","Text":"to 13 over 8 Xg prime."},{"Start":"11:16.555 ","End":"11:24.130","Text":"These are the 2 equations that I will need, and now all I have to do is get to"},{"Start":"11:24.130 ","End":"11:33.310","Text":"the heading substitute. I\u0027m going to substitute into this"},{"Start":"11:33.310 ","End":"11:37.420","Text":"one here, and there\u0027s something strange that happens because it turns"},{"Start":"11:37.420 ","End":"11:43.090","Text":"out that I have Xg prime that\u0027s from this given,"},{"Start":"11:43.090 ","End":"11:48.910","Text":"and I\u0027m looking for X prime but nowhere do I use the fact"},{"Start":"11:48.910 ","End":"11:55.045","Text":"that this Xg is 30 in both of these."},{"Start":"11:55.045 ","End":"11:58.420","Text":"This is something that\u0027s not used."},{"Start":"11:58.420 ","End":"12:02.110","Text":"Apparently, it doesn\u0027t depend on Xg and this can occasionally"},{"Start":"12:02.110 ","End":"12:07.190","Text":"happen where an answer doesn\u0027t depend on 1 of the things you\u0027re given."},{"Start":"12:12.150 ","End":"12:15.575","Text":"Where was it? Here, yeah."},{"Start":"12:15.575 ","End":"12:18.885","Text":"Underline this will find it better."},{"Start":"12:18.885 ","End":"12:22.380","Text":"X prime, which is what we\u0027re looking for,"},{"Start":"12:22.380 ","End":"12:25.785","Text":"when Xg is 30, only there\u0027s nowhere to substitute Xg,"},{"Start":"12:25.785 ","End":"12:28.070","Text":"so it must be everywhere,"},{"Start":"12:28.070 ","End":"12:37.900","Text":"is equal to 13 over 8, and Xg prime is from the given times 3."},{"Start":"12:37.900 ","End":"12:45.140","Text":"We just get that X prime is 39 over 8."},{"Start":"12:49.650 ","End":"12:53.200","Text":"I guess I should have said substitute"},{"Start":"12:53.200 ","End":"12:59.500","Text":"plus solve, I often throw these sections and once you start substituting,"},{"Start":"12:59.500 ","End":"13:02.029","Text":"you just can\u0027t help solving."},{"Start":"13:02.100 ","End":"13:08.380","Text":"The final thing to do is just to write this in readable language,"},{"Start":"13:08.380 ","End":"13:14.170","Text":"not mathematical, some statement of what we\u0027ve found and with the units and everything."},{"Start":"13:14.170 ","End":"13:17.360","Text":"It will go something like this."},{"Start":"13:18.180 ","End":"13:24.310","Text":"The tip of the shadow is moving away from the pole at a rate of"},{"Start":"13:24.310 ","End":"13:31.720","Text":"4.875 feet per second and I could add when the girl is 30-foot from the pole,"},{"Start":"13:31.720 ","End":"13:33.490","Text":"but we found that it wasn\u0027t used such"},{"Start":"13:33.490 ","End":"13:35.770","Text":"that whether we\u0027re even putting it in at the answer."},{"Start":"13:35.770 ","End":"13:37.570","Text":"That\u0027s part a we\u0027re not done."},{"Start":"13:37.570 ","End":"13:42.775","Text":"We still have part b. I just like to point out"},{"Start":"13:42.775 ","End":"13:46.630","Text":"the part b will become quite easy if we realize that"},{"Start":"13:46.630 ","End":"13:50.620","Text":"what we\u0027ve just found here or here was X prime,"},{"Start":"13:50.620 ","End":"13:59.800","Text":"and we have Xg prime. We also know that X prime is Xg prime plus X prime."},{"Start":"13:59.800 ","End":"14:05.750","Text":"That it just summarize those, and they\u0027ll go on to the next page."},{"Start":"14:07.290 ","End":"14:11.275","Text":"I brought all the important facts from the other page,"},{"Start":"14:11.275 ","End":"14:16.030","Text":"and we\u0027re in part b now. Remember we had this equation,"},{"Start":"14:16.030 ","End":"14:22.615","Text":"with the derivatives, and then we solved part a and got what X prime was."},{"Start":"14:22.615 ","End":"14:26.380","Text":"We also knew Xg prime from the given, and what we have to"},{"Start":"14:26.380 ","End":"14:30.550","Text":"find out in b is how fast the shadow is lengthening"},{"Start":"14:30.550 ","End":"14:35.480","Text":"or how quickly the tip of the shadow is moving away from the girl. All we"},{"Start":"14:35.480 ","End":"14:37.400","Text":"have to do is since we have this equation with"},{"Start":"14:37.400 ","End":"14:40.460","Text":"3 quantities and we\u0027re given 2, is just find the third."},{"Start":"14:40.460 ","End":"14:42.575","Text":"Probably we can do it in our heads."},{"Start":"14:42.575 ","End":"14:50.570","Text":"If we have the 4.875 equals 3 plus something, then that something is going to be 1.875."},{"Start":"14:50.570 ","End":"14:54.845","Text":"I\u0027ll just take out the eraser, erase this,"},{"Start":"14:54.845 ","End":"15:03.830","Text":"and instead of it put 1.875 feet per second and finally, as before,"},{"Start":"15:03.830 ","End":"15:09.560","Text":"we just write something in words like the tip of the shadow is moving"},{"Start":"15:09.560 ","End":"15:15.470","Text":"away from the girl at a rate of 1.875 feet per second."},{"Start":"15:15.470 ","End":"15:21.590","Text":"I should add when the girl is 30 foot away from the pole,"},{"Start":"15:21.590 ","End":"15:26.225","Text":"but turned out that didn\u0027t make any difference, so I\u0027m leaving it out."},{"Start":"15:26.225 ","End":"15:32.430","Text":"That\u0027s the end of this exercise finally."}],"ID":1467},{"Watched":false,"Name":"Exercise 7","Duration":"10m 3s","ChapterTopicVideoID":1455,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this exercise, we have a related rates problem,"},{"Start":"00:05.220 ","End":"00:07.740","Text":"and first thing we should do is to read it."},{"Start":"00:07.740 ","End":"00:13.470","Text":"A spotlight is on the ground 24 feet away from a wall and"},{"Start":"00:13.470 ","End":"00:19.650","Text":"a 4 foot tall girl is walking towards the wall at a rate of 2 feet per second."},{"Start":"00:19.650 ","End":"00:26.400","Text":"How fast is the height of the shadow changing when the girl is 10 feet from the wall?"},{"Start":"00:26.400 ","End":"00:31.690","Text":"Is the shadow increasing or decreasing in height at this time?"},{"Start":"00:32.440 ","End":"00:38.540","Text":"Fortunately, we are given a sketch or at least a partial sketch here."},{"Start":"00:38.540 ","End":"00:42.885","Text":"This heading will go together with the denote where I name variables,"},{"Start":"00:42.885 ","End":"00:44.375","Text":"the change over time,"},{"Start":"00:44.375 ","End":"00:46.390","Text":"and we\u0027ll do them together."},{"Start":"00:46.390 ","End":"00:49.420","Text":"Each will help me with the other."},{"Start":"00:49.880 ","End":"00:54.305","Text":"Let\u0027s start with filling in some stuff on the sketch."},{"Start":"00:54.305 ","End":"00:58.054","Text":"Now, the spotlight is on the ground 24 feet away from the wall,"},{"Start":"00:58.054 ","End":"01:04.410","Text":"which means that this is 24 and it\u0027s a constant and it\u0027s not changing with time."},{"Start":"01:05.000 ","End":"01:10.915","Text":"The other thing we have is that the girl is 4 feet tall."},{"Start":"01:10.915 ","End":"01:14.095","Text":"So here I\u0027ll write 4."},{"Start":"01:14.095 ","End":"01:18.920","Text":"In general, I won\u0027t bother with the units here because all the units of length will be in"},{"Start":"01:18.920 ","End":"01:23.990","Text":"feet and the units of time will be in seconds."},{"Start":"01:23.990 ","End":"01:28.490","Text":"For example, rates of change of distance will be speed,"},{"Start":"01:28.490 ","End":"01:30.970","Text":"will be feet per second, and so on."},{"Start":"01:30.970 ","End":"01:36.000","Text":"What we\u0027re looking for are variables that change with time."},{"Start":"01:36.000 ","End":"01:38.000","Text":"Let\u0027s see what we have."},{"Start":"01:38.000 ","End":"01:42.670","Text":"Let\u0027s call the height of the shadow of the girl,"},{"Start":"01:42.670 ","End":"01:46.810","Text":"y, we\u0027ll denote that by y."},{"Start":"01:47.300 ","End":"01:54.620","Text":"Let\u0027s take x as the distance from the girl to the spotlight."},{"Start":"01:54.620 ","End":"01:58.530","Text":"This will be x."},{"Start":"01:58.530 ","End":"02:02.360","Text":"We can fill in a few others that follow."},{"Start":"02:02.360 ","End":"02:08.820","Text":"For example, if this is x and this is going to be 24 minus x."},{"Start":"02:09.550 ","End":"02:16.105","Text":"The only other thing we know really is that the girl,"},{"Start":"02:16.105 ","End":"02:18.680","Text":"depends which way you look at it is either walking"},{"Start":"02:18.680 ","End":"02:21.125","Text":"towards the wall or away from the spotlight,"},{"Start":"02:21.125 ","End":"02:23.620","Text":"but either way, it\u0027s the same thing,"},{"Start":"02:23.620 ","End":"02:26.690","Text":"and it\u0027s at 2 feet a second."},{"Start":"02:26.690 ","End":"02:28.670","Text":"If we write those variables here,"},{"Start":"02:28.670 ","End":"02:32.840","Text":"what we have basically is x and y and their derivatives."},{"Start":"02:32.840 ","End":"02:40.095","Text":"We get that x is equal to and y is equal to,"},{"Start":"02:40.095 ","End":"02:43.800","Text":"and I\u0027ll write their definitions in a moment."},{"Start":"02:43.930 ","End":"02:50.795","Text":"So x is the distance from the spotlight to the girl and that\u0027s in feet,"},{"Start":"02:50.795 ","End":"02:55.325","Text":"and y is the height of the shadow, also in feet."},{"Start":"02:55.325 ","End":"02:59.265","Text":"In the next section or heading,"},{"Start":"02:59.265 ","End":"03:02.800","Text":"which I\u0027ll call hence."},{"Start":"03:02.960 ","End":"03:10.085","Text":"Under this heading, I\u0027m going to put the derivatives of these,"},{"Start":"03:10.085 ","End":"03:14.600","Text":"which is x prime and y prime, and of course,"},{"Start":"03:14.600 ","End":"03:17.570","Text":"these are going to be in feet per second,"},{"Start":"03:17.570 ","End":"03:19.620","Text":"and I\u0027m going to get them."},{"Start":"03:19.620 ","End":"03:24.365","Text":"These are just the derivatives of these with respect to time."},{"Start":"03:24.365 ","End":"03:28.270","Text":"Oh, I just should have mentioned on the diagram,"},{"Start":"03:28.270 ","End":"03:31.660","Text":"and you can see that this is x prime"},{"Start":"03:31.660 ","End":"03:37.235","Text":"because she\u0027s walking towards the wall or she\u0027s walking away from the spotlight,"},{"Start":"03:37.235 ","End":"03:39.520","Text":"that distance is exactly x."},{"Start":"03:39.520 ","End":"03:43.610","Text":"So x is increasing at 2 feet per second."},{"Start":"03:45.470 ","End":"03:50.260","Text":"So x prime is the rate of change of the distance from the spotlight to"},{"Start":"03:50.260 ","End":"03:54.805","Text":"the girl and y prime is the rate of change of the height of the shadow."},{"Start":"03:54.805 ","End":"04:03.160","Text":"I should have written in the units that they are both in feet per second."},{"Start":"04:06.440 ","End":"04:14.630","Text":"The next sections are going to be what we\u0027re looking for and what we\u0027re given."},{"Start":"04:15.800 ","End":"04:19.215","Text":"Here we are with the given and the looking for."},{"Start":"04:19.215 ","End":"04:25.880","Text":"Given is basically this quantity that x prime,"},{"Start":"04:26.360 ","End":"04:31.250","Text":"which again is a function of t but we\u0027ve stopped writing the bracket, t,"},{"Start":"04:31.250 ","End":"04:35.310","Text":"is equal to 2,"},{"Start":"04:35.310 ","End":"04:41.085","Text":"and that\u0027s for every t. What we\u0027re looking for is y prime."},{"Start":"04:41.085 ","End":"04:42.545","Text":"It\u0027s exactly what we want,"},{"Start":"04:42.545 ","End":"04:45.920","Text":"is the rate of change of the height of the shadow."},{"Start":"04:49.550 ","End":"04:51.695","Text":"It\u0027s said in the question,"},{"Start":"04:51.695 ","End":"04:57.245","Text":"when the girl is 10 feet away from the wall,"},{"Start":"04:57.245 ","End":"05:01.955","Text":"which is when 24 minus x,"},{"Start":"05:01.955 ","End":"05:04.160","Text":"which is this distance,"},{"Start":"05:04.160 ","End":"05:10.860","Text":"is 10 feet away from the wall."},{"Start":"05:11.350 ","End":"05:17.510","Text":"I suppose I could just write this right away as x equals 14."},{"Start":"05:17.510 ","End":"05:18.800","Text":"I\u0027ll just put it here."},{"Start":"05:18.800 ","End":"05:22.280","Text":"That means when x is equal to 14,"},{"Start":"05:22.280 ","End":"05:25.800","Text":"do the 24 minus 10 in our heads."},{"Start":"05:26.800 ","End":"05:31.475","Text":"Next, we need to find an equation or more,"},{"Start":"05:31.475 ","End":"05:36.470","Text":"1 or more that ties in all the variables that we\u0027ve found,"},{"Start":"05:36.470 ","End":"05:38.690","Text":"in this case, x and y,"},{"Start":"05:38.690 ","End":"05:41.180","Text":"and I call this equation."},{"Start":"05:41.180 ","End":"05:46.380","Text":"We\u0027re going to get this actually from similar triangles and nothing more."},{"Start":"05:46.380 ","End":"05:48.350","Text":"In previous exercises, we use"},{"Start":"05:48.350 ","End":"05:51.740","Text":"similar triangles as a way of eliminating an extra variable."},{"Start":"05:51.740 ","End":"05:53.885","Text":"Here, that\u0027s all we\u0027re going to need."},{"Start":"05:53.885 ","End":"05:57.080","Text":"Look, we have a similar triangle."},{"Start":"05:57.080 ","End":"06:00.065","Text":"We have the big triangle here,"},{"Start":"06:00.065 ","End":"06:04.804","Text":"which I\u0027ll highlight in red."},{"Start":"06:04.804 ","End":"06:07.735","Text":"That\u0027s 1 triangle."},{"Start":"06:07.735 ","End":"06:10.080","Text":"That\u0027s the 1 that stays fixed."},{"Start":"06:10.080 ","End":"06:12.980","Text":"Then this 1, which the girl forms with"},{"Start":"06:12.980 ","End":"06:18.245","Text":"the spotlight is another 1 and I\u0027ll do it in a different color, of course."},{"Start":"06:18.245 ","End":"06:24.740","Text":"The red 1 and the blue 1 are similar triangles and from this,"},{"Start":"06:24.740 ","End":"06:29.790","Text":"we can get that ratio of similar sides is the same."},{"Start":"06:33.350 ","End":"06:37.550","Text":"Let\u0027s just see, not too much room here. Yeah, here we are."},{"Start":"06:37.550 ","End":"06:41.465","Text":"If we take the shadow"},{"Start":"06:41.465 ","End":"06:49.215","Text":"over the distance 24,"},{"Start":"06:49.215 ","End":"06:51.390","Text":"and that\u0027s y over 24,"},{"Start":"06:51.390 ","End":"06:53.890","Text":"that\u0027s from the red triangle."},{"Start":"06:55.750 ","End":"06:58.370","Text":"Then from the blue triangle,"},{"Start":"06:58.370 ","End":"07:03.159","Text":"we\u0027ll get the height of the girl over x."},{"Start":"07:03.159 ","End":"07:10.190","Text":"The girl is 4, sorry, over x."},{"Start":"07:10.190 ","End":"07:18.920","Text":"What that gives us is that y equals 24 times 4,"},{"Start":"07:18.920 ","End":"07:24.155","Text":"which is 96 over x."},{"Start":"07:24.155 ","End":"07:26.720","Text":"Now when we have the equation,"},{"Start":"07:26.720 ","End":"07:30.560","Text":"it\u0027s almost always useful to differentiate it,"},{"Start":"07:30.560 ","End":"07:34.450","Text":"so we get the derived equation."},{"Start":"07:34.450 ","End":"07:43.770","Text":"Here we are. When we derive, then looking here,"},{"Start":"07:43.770 ","End":"07:48.845","Text":"that y prime is"},{"Start":"07:48.845 ","End":"07:55.580","Text":"minus 96 over x squared,"},{"Start":"07:55.580 ","End":"08:02.400","Text":"but times x prime because x is still a function of t. Next after this,"},{"Start":"08:02.400 ","End":"08:07.450","Text":"we get to substitute that heading."},{"Start":"08:08.360 ","End":"08:13.010","Text":"We have everything to substitute here except for y prime,"},{"Start":"08:13.010 ","End":"08:15.660","Text":"which is what we\u0027re looking for."},{"Start":"08:16.580 ","End":"08:23.345","Text":"That y prime is equal to minus 96."},{"Start":"08:23.345 ","End":"08:25.250","Text":"Now, x prime we get from here,"},{"Start":"08:25.250 ","End":"08:27.835","Text":"from the given which is 2,"},{"Start":"08:27.835 ","End":"08:32.179","Text":"and on the denominator, x squared."},{"Start":"08:32.179 ","End":"08:34.145","Text":"But we know what x is."},{"Start":"08:34.145 ","End":"08:38.165","Text":"We are given that x is 14,"},{"Start":"08:38.165 ","End":"08:40.940","Text":"so over 14 squared."},{"Start":"08:40.940 ","End":"08:44.735","Text":"This is now just an arithmetic problem."},{"Start":"08:44.735 ","End":"08:47.500","Text":"The answer comes out."},{"Start":"08:47.500 ","End":"08:55.895","Text":"That cancels to minus 48 over 49 and that is equal to approximately,"},{"Start":"08:55.895 ","End":"08:57.575","Text":"yeah, there it is,"},{"Start":"08:57.575 ","End":"09:04.905","Text":"minus 0.98, and that of course is in feet per second."},{"Start":"09:04.905 ","End":"09:09.770","Text":"It does answer also the question as to whether the shadow is lengthening or not."},{"Start":"09:09.770 ","End":"09:11.690","Text":"Since the derivative is negative,"},{"Start":"09:11.690 ","End":"09:14.420","Text":"the shadow is actually getting shorter."},{"Start":"09:14.420 ","End":"09:18.455","Text":"This is actually the answer except that at the end,"},{"Start":"09:18.455 ","End":"09:26.570","Text":"I like to write the answer in simple words and not in mathematical language."},{"Start":"09:26.570 ","End":"09:31.920","Text":"What we would say is that when the girl is,"},{"Start":"09:31.930 ","End":"09:35.900","Text":"what was it, 10 feet away from the wall,"},{"Start":"09:35.900 ","End":"09:41.300","Text":"that the height of"},{"Start":"09:41.300 ","End":"09:47.975","Text":"the shadow is decreasing at a rate of 0.98 feet per second."},{"Start":"09:47.975 ","End":"09:52.760","Text":"Notice I didn\u0027t just write changing when I know if it\u0027s positive or negative,"},{"Start":"09:52.760 ","End":"09:57.830","Text":"I\u0027d like to write increasing or decreasing as it is in this case."},{"Start":"09:58.880 ","End":"10:03.130","Text":"We\u0027re done for this exercise."}],"ID":1468},{"Watched":false,"Name":"Exercise 8","Duration":"14m 56s","ChapterTopicVideoID":1456,"CourseChapterTopicPlaylistID":1565,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"We have here a related rates problem and let\u0027s read it first."},{"Start":"00:05.730 ","End":"00:10.695","Text":"Brad and Angelina on bikes are separated by 500 meters."},{"Start":"00:10.695 ","End":"00:14.790","Text":"Brad starts riding north at a rate of 4 meters a second."},{"Start":"00:14.790 ","End":"00:20.610","Text":"6 minutes later, Angelina starts riding south at 4 meters a second."},{"Start":"00:20.610 ","End":"00:22.395","Text":"Just the same rate."},{"Start":"00:22.395 ","End":"00:26.205","Text":"At what rate is the distance separating Brad and Angelina"},{"Start":"00:26.205 ","End":"00:31.485","Text":"changing 25 minutes after Brad starts riding?"},{"Start":"00:31.485 ","End":"00:38.955","Text":"What I usually do is have a section heading on denoting variables."},{"Start":"00:38.955 ","End":"00:40.650","Text":"But there\u0027s also a sketch."},{"Start":"00:40.650 ","End":"00:42.335","Text":"We\u0027ll combine them both,"},{"Start":"00:42.335 ","End":"00:45.595","Text":"we\u0027ll complete the sketch."},{"Start":"00:45.595 ","End":"00:51.030","Text":"I\u0027m going to also denote certain quantities with variable names."},{"Start":"00:51.370 ","End":"01:01.300","Text":"Let\u0027s take a look at this sketch and let\u0027s see now."},{"Start":"01:01.300 ","End":"01:04.250","Text":"Well, the first variable,"},{"Start":"01:04.250 ","End":"01:07.985","Text":"we don\u0027t see its time itself."},{"Start":"01:07.985 ","End":"01:10.625","Text":"But let\u0027s say that time,"},{"Start":"01:10.625 ","End":"01:12.890","Text":"which will be denoted t,"},{"Start":"01:12.890 ","End":"01:19.430","Text":"will be in seconds because I see there\u0027s also minutes used here,"},{"Start":"01:19.430 ","End":"01:21.830","Text":"so we\u0027ll convert later the minutes to seconds."},{"Start":"01:21.830 ","End":"01:25.830","Text":"As for distance, the distance,"},{"Start":"01:25.830 ","End":"01:30.570","Text":"I see only 1 kind of unit and that\u0027s meters so we\u0027ll go with meters."},{"Start":"01:30.570 ","End":"01:33.170","Text":"I\u0027m just mentioning the units we\u0027re going to be used."},{"Start":"01:33.170 ","End":"01:36.875","Text":"Meters m, seconds sec,"},{"Start":"01:36.875 ","End":"01:42.020","Text":"and we have 1 distance here,"},{"Start":"01:42.020 ","End":"01:43.370","Text":"which is how far, this is,"},{"Start":"01:43.370 ","End":"01:45.110","Text":"I guess where Brad starts."},{"Start":"01:45.110 ","End":"01:50.455","Text":"He starts at this point here and he works his way upward."},{"Start":"01:50.455 ","End":"01:56.550","Text":"This is the point at which Brad is and he\u0027s covered x meters of distance."},{"Start":"01:56.550 ","End":"01:59.705","Text":"Angelina began here and traveled south."},{"Start":"01:59.705 ","End":"02:06.970","Text":"Let\u0027s say she traveled y also meters and the distance between them will be z."},{"Start":"02:06.970 ","End":"02:12.705","Text":"Basically, we have 3 units of distance in meters,"},{"Start":"02:12.705 ","End":"02:18.770","Text":"and speeds are measured in meters per second because we\u0027re using that."},{"Start":"02:18.770 ","End":"02:24.170","Text":"There are also various constants or at least there is 1 constant distance,"},{"Start":"02:24.170 ","End":"02:30.360","Text":"which is the distance between here and here."},{"Start":"02:30.360 ","End":"02:34.840","Text":"That\u0027s given to be 500 meters"},{"Start":"02:35.480 ","End":"02:40.940","Text":"and this is just the sum of the distances of"},{"Start":"02:40.940 ","End":"02:45.640","Text":"x and y and this will help us make a Pythagorean triangle mainly."},{"Start":"02:45.640 ","End":"02:48.870","Text":"This will be x plus y,"},{"Start":"02:48.870 ","End":"02:53.150","Text":"as you can see extra here and y from here and they\u0027re also"},{"Start":"02:53.150 ","End":"02:58.399","Text":"moving although they started off at different times, they\u0027re both moving."},{"Start":"02:58.399 ","End":"03:06.510","Text":"Brad is going north at the rate of x prime if you like,"},{"Start":"03:06.510 ","End":"03:10.785","Text":"which is going to be 4 meters a second,"},{"Start":"03:10.785 ","End":"03:14.415","Text":"4 meters per second."},{"Start":"03:14.415 ","End":"03:20.465","Text":"I suppose I could write this as meters but it\u0027ll get cluttered and then we have"},{"Start":"03:20.465 ","End":"03:29.680","Text":"Angelina who is going south and that would be y prime."},{"Start":"03:30.020 ","End":"03:37.185","Text":"I\u0027m measuring y as just a distance not in any particular direction or if you like,"},{"Start":"03:37.185 ","End":"03:41.390","Text":"I menti1d x is measured going north and y is measured going south."},{"Start":"03:41.390 ","End":"03:51.615","Text":"Both the speeds are positive and y prime is also 4 meters per second."},{"Start":"03:51.615 ","End":"03:55.130","Text":"That\u0027s basically it for the sketch denote part."},{"Start":"03:55.130 ","End":"03:59.900","Text":"Now usually I have a thing where I give an explanation for each variable,"},{"Start":"03:59.900 ","End":"04:02.270","Text":"but there\u0027s no need to do that."},{"Start":"04:02.270 ","End":"04:03.860","Text":"The picture is fairly explanatory,"},{"Start":"04:03.860 ","End":"04:05.870","Text":"I\u0027ll just say that we have x,"},{"Start":"04:05.870 ","End":"04:10.320","Text":"y, and z, are 3 distances."},{"Start":"04:11.590 ","End":"04:21.230","Text":"They\u0027re all in meters and if we derive these,"},{"Start":"04:21.230 ","End":"04:24.080","Text":"we also have x prime,"},{"Start":"04:24.080 ","End":"04:27.560","Text":"y prime, and z prime,"},{"Start":"04:27.560 ","End":"04:31.620","Text":"which are the rates of change of these."},{"Start":"04:34.550 ","End":"04:36.570","Text":"That\u0027s the rate of change,"},{"Start":"04:36.570 ","End":"04:39.040","Text":"it\u0027s a derivative with respect to time."},{"Start":"04:39.040 ","End":"04:48.270","Text":"The rates of change of these 3 and these are in meters per second."},{"Start":"04:48.270 ","End":"04:51.790","Text":"That\u0027s basically all I need to write about them,"},{"Start":"04:51.790 ","End":"04:54.565","Text":"that together with the picture makes everything clear."},{"Start":"04:54.565 ","End":"05:02.875","Text":"The next 2 headings are what we\u0027re given and what we\u0027re looking for."},{"Start":"05:02.875 ","End":"05:10.735","Text":"I\u0027ll write this 1 as given and then we\u0027ll have looking for,"},{"Start":"05:10.735 ","End":"05:18.385","Text":"and we\u0027re given basically that x prime is 4,"},{"Start":"05:18.385 ","End":"05:22.355","Text":"and also that y prime is equal to 4."},{"Start":"05:22.355 ","End":"05:28.744","Text":"But we\u0027re also given various miscellaneous initial data."},{"Start":"05:28.744 ","End":"05:34.010","Text":"I won\u0027t count that like the 500 and so on,"},{"Start":"05:34.360 ","End":"05:38.830","Text":"sometimes I think I should write plus initial data,"},{"Start":"05:38.830 ","End":"05:41.750","Text":"just the constants, and so forth."},{"Start":"05:41.750 ","End":"05:46.910","Text":"These are the main things relating to the variables that change with time."},{"Start":"05:46.910 ","End":"05:50.300","Text":"Now, what we\u0027re looking for is"},{"Start":"05:50.300 ","End":"05:55.100","Text":"the way the distance between Brad and Angelina changes, and that\u0027s z."},{"Start":"05:55.100 ","End":"06:02.240","Text":"What we\u0027re looking for is z prime but at a certain moment and"},{"Start":"06:02.240 ","End":"06:11.015","Text":"that\u0027s at the time that Brad has ridden for 25 minutes."},{"Start":"06:11.015 ","End":"06:13.525","Text":"That\u0027s the particular moment in time."},{"Start":"06:13.525 ","End":"06:20.645","Text":"Next, we\u0027re going to need 1 or more equations that tie in the variables together."},{"Start":"06:20.645 ","End":"06:25.760","Text":"Let me just call that section equation."},{"Start":"06:25.760 ","End":"06:29.220","Text":"Well, possibly more than 1,"},{"Start":"06:29.220 ","End":"06:33.315","Text":"there\u0027s 1 basic 1 and then there\u0027s again all the initial stuff."},{"Start":"06:33.315 ","End":"06:38.195","Text":"The main equation is just Pythagoras\u0027s theorem."},{"Start":"06:38.195 ","End":"06:46.790","Text":"I mean here we have a right-angled triangle and what we can say is that the diagonal,"},{"Start":"06:46.790 ","End":"06:51.820","Text":"the hypotenuse really is z squared."},{"Start":"06:51.820 ","End":"06:54.525","Text":"We\u0027re all talking still in meters now,"},{"Start":"06:54.525 ","End":"06:57.959","Text":"is equal to the length of the 1 side,"},{"Start":"06:57.959 ","End":"07:03.890","Text":"which is x plus y squared,"},{"Start":"07:03.890 ","End":"07:08.810","Text":"and the other side, that\u0027s 500 squared."},{"Start":"07:08.810 ","End":"07:13.085","Text":"That\u0027s the 1 equation and we also have,"},{"Start":"07:13.085 ","End":"07:16.340","Text":"I\u0027d like to mention it because there\u0027s all sorts of"},{"Start":"07:16.340 ","End":"07:19.940","Text":"initial data which are like an equation like the time"},{"Start":"07:19.940 ","End":"07:27.675","Text":"that Brad\u0027s been riding is 6 minutes longer than Angelina\u0027s been riding and so on,"},{"Start":"07:27.675 ","End":"07:32.670","Text":"so plus initial data."},{"Start":"07:32.670 ","End":"07:35.830","Text":"That 1 was riding North, that 1 was riding South."},{"Start":"07:35.830 ","End":"07:37.270","Text":"All these initial stuff,"},{"Start":"07:37.270 ","End":"07:39.955","Text":"it\u0027s like an equation on these values,"},{"Start":"07:39.955 ","End":"07:43.750","Text":"but this is the main equation that we\u0027re going to use for relating"},{"Start":"07:43.750 ","End":"07:49.060","Text":"the derivatives and this will be for the related rates."},{"Start":"07:49.060 ","End":"07:51.715","Text":"That\u0027s the equation."},{"Start":"07:51.715 ","End":"07:53.950","Text":"After the equation,"},{"Start":"07:53.950 ","End":"07:56.380","Text":"I like to usually derive it,"},{"Start":"07:56.380 ","End":"07:59.180","Text":"it always gives us something."},{"Start":"08:01.500 ","End":"08:07.870","Text":"We take the derivative and the derivative is with respect to time."},{"Start":"08:07.870 ","End":"08:10.090","Text":"We have to remember that all these are functions of time,"},{"Start":"08:10.090 ","End":"08:15.160","Text":"we just stopped writing the parentheses t all the time being a nuisance."},{"Start":"08:15.160 ","End":"08:22.165","Text":"We get the derivative of z squared is 2z z prime,"},{"Start":"08:22.165 ","End":"08:25.345","Text":"and I forgot in the United States we say z and not z."},{"Start":"08:25.345 ","End":"08:27.520","Text":"Some people cross their zs,"},{"Start":"08:27.520 ","End":"08:30.700","Text":"some people don\u0027t, just make a note of that."},{"Start":"08:30.700 ","End":"08:35.005","Text":"Some people like to write their zs like this and they call it z,"},{"Start":"08:35.005 ","End":"08:37.765","Text":"other people write it with a little,"},{"Start":"08:37.765 ","End":"08:40.015","Text":"this is more the of European style."},{"Start":"08:40.015 ","End":"08:42.940","Text":"I don\u0027t want you to get confused by a notation."},{"Start":"08:42.940 ","End":"08:46.060","Text":"This is just a notation and the way we pronounce things,"},{"Start":"08:46.060 ","End":"08:49.240","Text":"it\u0027s the 1 letter that the English and American say different."},{"Start":"08:49.240 ","End":"08:51.670","Text":"In England z, in America z,"},{"Start":"08:51.670 ","End":"08:53.695","Text":"excuse me, which any 1 I say."},{"Start":"08:53.695 ","End":"09:02.320","Text":"So 2z z prime is equal to 2,"},{"Start":"09:02.320 ","End":"09:07.420","Text":"x plus y, and x plus y all prime,"},{"Start":"09:07.420 ","End":"09:11.320","Text":"and I can put the prime in separately in each 1."},{"Start":"09:11.320 ","End":"09:14.650","Text":"The 500 is just a constant,"},{"Start":"09:14.650 ","End":"09:16.030","Text":"so it doesn\u0027t go anywhere."},{"Start":"09:16.030 ","End":"09:17.740","Text":"This is what we get and of course,"},{"Start":"09:17.740 ","End":"09:22.330","Text":"we could cancel the 2 which I\u0027m tempted to do right away."},{"Start":"09:22.330 ","End":"09:25.945","Text":"There\u0027s so much stuff we don\u0027t know here,"},{"Start":"09:25.945 ","End":"09:27.700","Text":"but we could easily compute."},{"Start":"09:27.700 ","End":"09:29.890","Text":"I\u0027m going to put a special heading in for"},{"Start":"09:29.890 ","End":"09:33.250","Text":"this exercise which I\u0027ll call initial computation."},{"Start":"09:33.250 ","End":"09:39.040","Text":"It\u0027s working out all these at the special time 25 minutes after Brad has ridden,"},{"Start":"09:39.040 ","End":"09:44.590","Text":"what is x and what is y and what is z and so on."},{"Start":"09:44.870 ","End":"09:49.750","Text":"I\u0027ll just call it initial computation."},{"Start":"09:49.760 ","End":"09:58.150","Text":"I\u0027ll do some conversions from minutes to seconds and so on and compute time differences."},{"Start":"10:00.920 ","End":"10:04.930","Text":"I need a bit of space here."},{"Start":"10:04.930 ","End":"10:10.780","Text":"I just added a note that with all these computations are all for the same moment in time,"},{"Start":"10:10.780 ","End":"10:14.020","Text":"which is after Brad has ridden 25 minutes."},{"Start":"10:14.020 ","End":"10:18.145","Text":"So x is the distance that Brad has traveled."},{"Start":"10:18.145 ","End":"10:22.915","Text":"He\u0027s been traveling for 25 minutes,"},{"Start":"10:22.915 ","End":"10:29.320","Text":"but there are 60 seconds in a minute because we\u0027re working in seconds,"},{"Start":"10:29.320 ","End":"10:36.380","Text":"and we have to multiply by the rate in meters per second, which is 4."},{"Start":"10:37.410 ","End":"10:42.835","Text":"This gives us 4 times 25 is"},{"Start":"10:42.835 ","End":"10:49.885","Text":"100 times 60 is 6,000 and the units are meters."},{"Start":"10:49.885 ","End":"10:54.970","Text":"Now, y is at that same moment,"},{"Start":"10:54.970 ","End":"10:59.630","Text":"how far has Angelina traveled."},{"Start":"10:59.760 ","End":"11:08.845","Text":"This comes out to be not 25 because she started 6 minutes after him, but 19."},{"Start":"11:08.845 ","End":"11:10.720","Text":"I\u0027m just going to do that in my head,"},{"Start":"11:10.720 ","End":"11:15.490","Text":"I\u0027m not going to say, 25 minus 6 is 19,"},{"Start":"11:15.490 ","End":"11:17.545","Text":"that\u0027s just a 6-minute delay."},{"Start":"11:17.545 ","End":"11:21.085","Text":"Again, it\u0027s minutes, we need to convert to seconds."},{"Start":"11:21.085 ","End":"11:24.295","Text":"That\u0027s how long she\u0027s been traveling in seconds,"},{"Start":"11:24.295 ","End":"11:28.600","Text":"times her rate in meters per seconds which was also 4 by coincidence."},{"Start":"11:28.600 ","End":"11:31.370","Text":"This is equal to,"},{"Start":"11:32.220 ","End":"11:36.590","Text":"I make it 4,560,"},{"Start":"11:38.640 ","End":"11:42.055","Text":"actually hand computed this,"},{"Start":"11:42.055 ","End":"11:47.530","Text":"meters and that gives us y."},{"Start":"11:47.530 ","End":"11:50.990","Text":"Next we need z."},{"Start":"11:56.010 ","End":"11:59.020","Text":"Here is the formula,"},{"Start":"11:59.020 ","End":"12:01.570","Text":"well, the square root of this formula."},{"Start":"12:01.570 ","End":"12:08.965","Text":"We need 1 big square root sign and we have x plus y all squared."},{"Start":"12:08.965 ","End":"12:18.740","Text":"So let\u0027s see, x plus y will be 6 and 4 is 10,560,"},{"Start":"12:18.740 ","End":"12:24.130","Text":"that\u0027s x plus y, all this squared plus the"},{"Start":"12:24.130 ","End":"12:30.520","Text":"500 squared and the square root of this."},{"Start":"12:30.520 ","End":"12:33.430","Text":"I compute that to"},{"Start":"12:33.430 ","End":"12:43.210","Text":"be 10,571.83 meters with"},{"Start":"12:43.210 ","End":"12:45.650","Text":"the help of a calculator."},{"Start":"12:46.200 ","End":"12:50.185","Text":"Now, it\u0027s time to do some substitutions."},{"Start":"12:50.185 ","End":"12:58.720","Text":"Let\u0027s recall this equation here which was the 1 we derived from the Pythagoras."},{"Start":"12:58.720 ","End":"13:01.420","Text":"If you look at it, basically,"},{"Start":"13:01.420 ","End":"13:04.150","Text":"we have everything at the given time."},{"Start":"13:04.150 ","End":"13:13.490","Text":"So let\u0027s just go and substitute the values z which is this here,"},{"Start":"13:19.680 ","End":"13:28.165","Text":"10,571 times z prime which was what we\u0027re looking for,"},{"Start":"13:28.165 ","End":"13:38.500","Text":"is equal to x plus y which is what we had here already under the square sign,"},{"Start":"13:38.500 ","End":"13:49.060","Text":"is 10,560 and x prime plus y prime."},{"Start":"13:49.060 ","End":"13:55.240","Text":"Both these rates of change are constant and each is equal to 4 meters per seconds,"},{"Start":"13:55.240 ","End":"13:57.610","Text":"so we have 4 plus 4,"},{"Start":"13:57.610 ","End":"14:00.980","Text":"1 for Brad, 1 for Angelina."},{"Start":"14:01.530 ","End":"14:04.090","Text":"Next thing we have to do,"},{"Start":"14:04.090 ","End":"14:11.125","Text":"although I wouldn\u0027t really give it a special heading is just to solve."},{"Start":"14:11.125 ","End":"14:16.495","Text":"We get z prime is equal to,"},{"Start":"14:16.495 ","End":"14:21.655","Text":"and if I compute this again on the calculator,"},{"Start":"14:21.655 ","End":"14:24.790","Text":"what I get equals approximately"},{"Start":"14:24.790 ","End":"14:32.930","Text":"7.991 meters per second."},{"Start":"14:33.000 ","End":"14:40.675","Text":"We\u0027re basically d1 except that we usually write some sentence in words."},{"Start":"14:40.675 ","End":"14:45.400","Text":"Thus, after Brad has ridden for 25 minutes,"},{"Start":"14:45.400 ","End":"14:54.265","Text":"he and Angelina are separating at a rate of 7.991 approximately meters per second."},{"Start":"14:54.265 ","End":"14:56.630","Text":"That\u0027s it. We\u0027re d1."}],"ID":1469}],"Thumbnail":null,"ID":1565}]

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1.1

1