WEBVTT
1
00:00:01.240 --> 00:00:05.280 A:middle L:90%
Okay. Okay, we have this triangle. I'm
2
00:00:05.280 --> 00:00:10.810 A:middle L:90%
going to draw it down here. Hope that's not
3
00:00:10.810 --> 00:00:14.750 A:middle L:90%
a triangle. Okay, here we go. And
4
00:00:14.750 --> 00:00:16.879 A:middle L:90%
let's say this side is A and this site is
5
00:00:16.879 --> 00:00:19.160 A:middle L:90%
B. And then theta is in between them.
6
00:00:19.640 --> 00:00:25.059 A:middle L:90%
Okay, In the first question part A A and
7
00:00:25.059 --> 00:00:27.980 A:middle L:90%
B are fixed. They are two and 3 and
8
00:00:27.980 --> 00:00:31.809 A:middle L:90%
there are not getting any bigger. And D Theta
9
00:00:31.809 --> 00:00:35.619 A:middle L:90%
DT, r Theta is changing the rate that data
10
00:00:35.619 --> 00:00:39.009 A:middle L:90%
changes is 0.2 radiance per minute. So since it's
11
00:00:39.009 --> 00:00:41.659 A:middle L:90%
a raid, its name is D. Theta DT
12
00:00:42.439 --> 00:00:47.439 A:middle L:90%
. Yeah. Okay. So this would be see
13
00:00:47.450 --> 00:00:51.159 A:middle L:90%
right here And then a was two and B was
14
00:00:51.159 --> 00:00:55.350 A:middle L:90%
three can update is getting bigger. Okay, so
15
00:00:55.350 --> 00:01:00.520 A:middle L:90%
in a little while it's this big and then in
16
00:01:00.530 --> 00:01:04.739 A:middle L:90%
another little while it's this big. Then in another
17
00:01:04.739 --> 00:01:07.790 A:middle L:90%
little while it's this big. But A and B
18
00:01:07.790 --> 00:01:10.829 A:middle L:90%
are staying the same length. But you can see
19
00:01:10.829 --> 00:01:12.599 A:middle L:90%
that the area is changing. Okay, so the
20
00:01:12.599 --> 00:01:15.049 A:middle L:90%
question is, how fast is the area change?
21
00:01:15.939 --> 00:01:19.129 A:middle L:90%
Okay, so we have a cause one half a
22
00:01:19.129 --> 00:01:25.659 A:middle L:90%
B signed data And a was to be was three
23
00:01:26.939 --> 00:01:32.040 A:middle L:90%
And d. Theta DT was.2. Okay,
24
00:01:32.040 --> 00:01:33.609 A:middle L:90%
so A and B are fixed. They're not going
25
00:01:33.609 --> 00:01:34.480 A:middle L:90%
to change. So let's put them in before we
26
00:01:34.480 --> 00:01:40.010 A:middle L:90%
take the derivative one half times two times three times
27
00:01:40.010 --> 00:01:44.390 A:middle L:90%
to sign up data. So a is three sine
28
00:01:44.390 --> 00:01:48.859 A:middle L:90%
theta. Okay, now the two things that are
29
00:01:48.859 --> 00:01:51.269 A:middle L:90%
changing here, R Theta and A. So when
30
00:01:51.269 --> 00:01:53.370 A:middle L:90%
we take their derivatives, they get A D T
31
00:01:53.370 --> 00:01:55.400 A:middle L:90%
stuck on it. So here we go derivative of
32
00:01:55.400 --> 00:01:59.730 A:middle L:90%
A D A D T. Derivative of three.
33
00:01:59.730 --> 00:02:04.329 A:middle L:90%
Sine theta is three. Cosign Theta times the derivative
34
00:02:04.329 --> 00:02:07.780 A:middle L:90%
of theta which is D theta DT. I think
35
00:02:07.780 --> 00:02:12.719 A:middle L:90%
I missed something here. Also find the ATT1 thing
36
00:02:12.719 --> 00:02:16.770 A:middle L:90%
. It is five or 3 fine. B A
37
00:02:16.770 --> 00:02:23.550 A:middle L:90%
T. T. Okay. So it's not saying
38
00:02:23.550 --> 00:02:25.479 A:middle L:90%
that data is pi over three all the time.
39
00:02:25.490 --> 00:02:29.740 A:middle L:90%
It says at one of these instances over here,
40
00:02:29.750 --> 00:02:31.479 A:middle L:90%
data is pi over three. And at that instant
41
00:02:31.490 --> 00:02:35.139 A:middle L:90%
, how fast is the area changing? Okay,
42
00:02:35.139 --> 00:02:38.939 A:middle L:90%
so just plug them in three Times The coastline of
43
00:02:38.939 --> 00:02:42.949 A:middle L:90%
pirate. three times D. Theta DT which was
44
00:02:43.340 --> 00:02:51.330 A:middle L:90%
0.2 543. That's 60°. That square 23 to 1
45
00:02:51.330 --> 00:02:53.780 A:middle L:90%
. So the coastline is one half. So three
46
00:02:53.780 --> 00:03:00.400 A:middle L:90%
times 1/2 times two. So three three what?
47
00:03:00.400 --> 00:03:02.659 A:middle L:90%
Well A and B. Are centimeters so it's centimeters
48
00:03:02.659 --> 00:03:07.680 A:middle L:90%
squared. And D Theta DT was in ratings per
49
00:03:07.680 --> 00:03:15.969 A:middle L:90%
minute. So it's centimeter squared per minute. Okay
50
00:03:15.979 --> 00:03:23.560 A:middle L:90%
? Now and the next one B is changing and
51
00:03:23.569 --> 00:03:27.030 A:middle L:90%
feta is changing. So the rate that B is
52
00:03:27.030 --> 00:03:29.129 A:middle L:90%
changing is called D. B D. T.
53
00:03:29.139 --> 00:03:30.210 A:middle L:90%
And the rate that that is changing is called D
54
00:03:30.210 --> 00:03:34.460 A:middle L:90%
. Theta DT but A is fixed this time.
55
00:03:35.539 --> 00:03:43.770 A:middle L:90%
Okay, So here's the picture. Okay. Is
56
00:03:43.770 --> 00:03:46.759 A:middle L:90%
to and then here's B. And here's data.
57
00:03:47.639 --> 00:03:52.370 A:middle L:90%
So here's our other side. All right, so
58
00:03:52.370 --> 00:03:55.659 A:middle L:90%
that is getting bigger. It is getting bigger.
59
00:03:57.740 --> 00:04:00.340 A:middle L:90%
See it getting bigger at the same time. Be
60
00:04:00.340 --> 00:04:03.409 A:middle L:90%
is getting bigger. So now we have this and
61
00:04:03.409 --> 00:04:06.719 A:middle L:90%
then tha tha gets even bigger and bigger gets even
62
00:04:06.719 --> 00:04:11.379 A:middle L:90%
bigger and now we have this and then they gets
63
00:04:11.379 --> 00:04:15.069 A:middle L:90%
even bigger and be gets even bigger and now we
64
00:04:15.069 --> 00:04:17.579 A:middle L:90%
have this. Okay, so two things are changing
65
00:04:17.579 --> 00:04:21.449 A:middle L:90%
this time be and theta, so A is fixed
66
00:04:21.449 --> 00:04:26.639 A:middle L:90%
at two B. Is changing at a rate of
67
00:04:26.649 --> 00:04:35.089 A:middle L:90%
, I forgot um 1.5 centimeters per minute and that
68
00:04:35.089 --> 00:04:41.050 A:middle L:90%
is changing at a rate of.2 radiance per minute
69
00:04:43.240 --> 00:04:45.930 A:middle L:90%
. And so we're trying to find D. D
70
00:04:45.930 --> 00:04:46.750 A:middle L:90%
. C. To D A D. T.
71
00:04:46.759 --> 00:04:54.949 A:middle L:90%
The rate the area's changing Win B is three and
72
00:04:54.949 --> 00:04:59.160 A:middle L:90%
theater is pi over three because 3 85 or three
73
00:05:00.040 --> 00:05:02.370 A:middle L:90%
. So it's like this is happening and we take
74
00:05:02.370 --> 00:05:04.699 A:middle L:90%
its picture right at the instant when B is three
75
00:05:04.699 --> 00:05:09.480 A:middle L:90%
and theta pi over three, what's the rate the
76
00:05:09.480 --> 00:05:13.250 A:middle L:90%
area is changing right there? Okay, same formula
77
00:05:13.839 --> 00:05:17.459 A:middle L:90%
A equals 1/2 a b. sine. Theta.
78
00:05:18.740 --> 00:05:21.129 A:middle L:90%
We can plug in equals two but we can't plug
79
00:05:21.129 --> 00:05:24.410 A:middle L:90%
in B equals three. Cosby is changing. So
80
00:05:24.410 --> 00:05:29.220 A:middle L:90%
you can't put anything in below this line until after
81
00:05:29.220 --> 00:05:32.339 A:middle L:90%
you take the derivative. Okay, so right before
82
00:05:32.339 --> 00:05:34.949 A:middle L:90%
you say fine you can't plug any of that other
83
00:05:34.949 --> 00:05:38.009 A:middle L:90%
stuff in but we can't play the two in before
84
00:05:38.009 --> 00:05:41.050 A:middle L:90%
we start. So we get one half times two
85
00:05:41.060 --> 00:05:45.759 A:middle L:90%
times B times signed data. So that's be santa
86
00:05:47.040 --> 00:05:49.129 A:middle L:90%
. Okay, now this time three things were changing
87
00:05:49.129 --> 00:05:53.290 A:middle L:90%
. Remember B is changing data is changing and the
88
00:05:53.290 --> 00:05:56.120 A:middle L:90%
area is changing. So over here on the right
89
00:05:56.129 --> 00:05:57.459 A:middle L:90%
, we're gonna have to use the product group.
90
00:05:58.339 --> 00:06:00.250 A:middle L:90%
Alright, here we go. The derivative of A
91
00:06:00.259 --> 00:06:01.649 A:middle L:90%
. Which is D. A. D. T
92
00:06:02.339 --> 00:06:05.730 A:middle L:90%
. Equals the first B. Times the derivative of
93
00:06:05.730 --> 00:06:10.949 A:middle L:90%
the second which is cosign Theta times the derivative of
94
00:06:10.949 --> 00:06:16.879 A:middle L:90%
theta plus the second to sync data times the derivative
95
00:06:16.879 --> 00:06:18.579 A:middle L:90%
of the first, which is D. B.
96
00:06:18.579 --> 00:06:25.240 A:middle L:90%
D. T. Okay, the first B derivative
97
00:06:25.240 --> 00:06:28.459 A:middle L:90%
of the second cosign Theta. D. Theta DT
98
00:06:28.470 --> 00:06:31.680 A:middle L:90%
plus the second santa times the derivative of the first
99
00:06:31.689 --> 00:06:34.379 A:middle L:90%
DVD T. Okay, so now just plug stuff
100
00:06:34.379 --> 00:06:39.250 A:middle L:90%
in. So b. is three Date is Pi
101
00:06:39.250 --> 00:06:44.680 A:middle L:90%
over three. D. Theta DT is.2 state
102
00:06:44.680 --> 00:06:46.300 A:middle L:90%
is five or three and D. V. D
103
00:06:46.300 --> 00:06:56.949 A:middle L:90%
. T. Was 1.5 543. Remember 60°2321. So
104
00:06:56.949 --> 00:07:00.889 A:middle L:90%
here we get three times 2 Times The Co Sign
105
00:07:00.889 --> 00:07:04.120 A:middle L:90%
, which is 1/2. Let's say we're here.
106
00:07:04.120 --> 00:07:09.810 A:middle L:90%
We get 1.5 Times The Sine which is the square
107
00:07:09.810 --> 00:07:15.860 A:middle L:90%
root of three over to. So it's three plus
108
00:07:15.439 --> 00:07:18.319 A:middle L:90%
.75 times the square to three. I don't know
109
00:07:18.319 --> 00:07:21.670 A:middle L:90%
where my calculator is, which is something and again
110
00:07:21.670 --> 00:07:26.959 A:middle L:90%
, cm squared per minute because it's area changing.
111
00:07:29.040 --> 00:07:31.100 A:middle L:90%
All right. Now in the 3rd 1 A.
112
00:07:31.100 --> 00:07:35.420 A:middle L:90%
Is changing B is changing data is changing. So
113
00:07:35.420 --> 00:07:40.790 A:middle L:90%
they're all getting bigger and they just keep getting bigger
114
00:07:40.819 --> 00:07:43.160 A:middle L:90%
and then let's find the A. D. T
115
00:07:44.139 --> 00:07:49.680 A:middle L:90%
. All right. So, we have D.
116
00:07:49.680 --> 00:07:53.990 A:middle L:90%
A. D. T. Was something D.
117
00:07:53.990 --> 00:07:57.220 A:middle L:90%
B. D. T. Was something D.
118
00:07:57.220 --> 00:08:03.319 A:middle L:90%
Theta DT was something And we want to find the
119
00:08:03.319 --> 00:08:05.569 A:middle L:90%
A. D. T. Wind. They have
120
00:08:05.569 --> 00:08:07.850 A:middle L:90%
to tell us what A. Is what B.
121
00:08:07.850 --> 00:08:13.060 A:middle L:90%
Is and what they do is can remember nothing below
122
00:08:13.060 --> 00:08:15.180 A:middle L:90%
this line. Can go into the equation until after
123
00:08:15.180 --> 00:08:16.689 A:middle L:90%
you take the derivative. So the A. D
124
00:08:16.689 --> 00:08:20.689 A:middle L:90%
. T. Was 1.51 point 5.2. Okay.
125
00:08:20.699 --> 00:08:26.589 A:middle L:90%
1.5. 1.52. And I bet this is 2
126
00:08:26.589 --> 00:08:35.990 A:middle L:90%
, 3 and five or 3. Okay, so
127
00:08:35.990 --> 00:08:41.000 A:middle L:90%
big A. Is one half a B. Sign
128
00:08:41.000 --> 00:08:45.000 A:middle L:90%
theater. Okay. This time we have four things
129
00:08:45.000 --> 00:08:48.200 A:middle L:90%
changing. Sorry is changing. It is changing state
130
00:08:48.200 --> 00:08:52.809 A:middle L:90%
is changing therefore the area is changing. So when
131
00:08:52.809 --> 00:08:54.090 A:middle L:90%
we take the derivative over here, on the right
132
00:08:54.090 --> 00:08:56.759 A:middle L:90%
hand side, we got three pieces. So we're
133
00:08:56.759 --> 00:09:00.710 A:middle L:90%
going to have to do a fancy fancy product rule
134
00:09:00.720 --> 00:09:01.350 A:middle L:90%
. So, let me show you an easy way
135
00:09:05.240 --> 00:09:07.990 A:middle L:90%
if why is the function F. G. H
136
00:09:09.000 --> 00:09:13.549 A:middle L:90%
. So three functions distributive is take the derivative of
137
00:09:13.559 --> 00:09:18.159 A:middle L:90%
F. Times G. H. Plus take the
138
00:09:18.159 --> 00:09:24.350 A:middle L:90%
derivative of G times FH plus the derivative of H
139
00:09:24.840 --> 00:09:26.419 A:middle L:90%
. Times the other two. So the derivative of
140
00:09:26.419 --> 00:09:30.450 A:middle L:90%
each one times the other two. That way we
141
00:09:30.450 --> 00:09:31.779 A:middle L:90%
don't have to do anything fancy. Alright, here
142
00:09:31.779 --> 00:09:35.059 A:middle L:90%
we go. D. A. D. T
143
00:09:35.740 --> 00:09:37.059 A:middle L:90%
. Equals. Okay, the one half is just
144
00:09:37.059 --> 00:09:39.059 A:middle L:90%
gonna be out in the front there. Okay,
145
00:09:39.059 --> 00:09:41.460 A:middle L:90%
the derivative of the first thing that's D. A
146
00:09:41.460 --> 00:09:48.070 A:middle L:90%
. D. T. Times the other two plus
147
00:09:48.340 --> 00:09:52.710 A:middle L:90%
the derivative of the middle thing. That's DBT Times
148
00:09:52.710 --> 00:09:58.639 A:middle L:90%
the other two plus the derivative of the third thing
149
00:09:58.649 --> 00:10:03.610 A:middle L:90%
which is coastline, tha tha d theta DT Times
150
00:10:03.610 --> 00:10:07.710 A:middle L:90%
The Other two Things. Okay in this rule work
151
00:10:07.710 --> 00:10:09.690 A:middle L:90%
, if you have four things or five things are
152
00:10:09.700 --> 00:10:11.879 A:middle L:90%
six things you take the derivative of one times the
153
00:10:11.879 --> 00:10:15.600 A:middle L:90%
other ones plus the derivative of the next one times
154
00:10:15.600 --> 00:10:18.000 A:middle L:90%
the other one. So you never have to use
155
00:10:18.000 --> 00:10:20.000 A:middle L:90%
the product rule more than once. All right now
156
00:10:20.000 --> 00:10:22.769 A:middle L:90%
we're gonna do is plug some stuff in. Okay
157
00:10:22.769 --> 00:10:28.210 A:middle L:90%
? We got 1.5 times three times a sign of
158
00:10:28.220 --> 00:10:35.169 A:middle L:90%
data which remember was squared 3/2 plus 1.5 times two
159
00:10:35.179 --> 00:10:39.179 A:middle L:90%
times the square to 3/2 Plus the co sign which
160
00:10:39.179 --> 00:10:43.350 A:middle L:90%
is 1/2 times point to Times two Times 3.
161
00:10:46.840 --> 00:10:48.490 A:middle L:90%
All right, let's see if I can do this
162
00:10:48.490 --> 00:10:54.600 A:middle L:90%
without a calculator. No. No. Okay.
163
00:10:54.600 --> 00:10:58.129 A:middle L:90%
So just calculate all of this. And then put
164
00:10:58.129 --> 00:11:01.419 A:middle L:90%
it put I mean put it in your calculator and
165
00:11:01.419 --> 00:11:03.059 A:middle L:90%
calculate and out will come the answer. I can
166
00:11:03.059 --> 00:11:05.480 A:middle L:90%
set it up a little bit more. Um Three
167
00:11:05.480 --> 00:11:09.399 A:middle L:90%
times 1.5. That's 4.5 divided by two. That's
168
00:11:09.399 --> 00:11:16.779 A:middle L:90%
2.25 square 23 plus. This one is 1.5 square
169
00:11:16.779 --> 00:11:24.440 A:middle L:90%
+23 Plus This one is um Uh those cancelled three
170
00:11:24.440 --> 00:11:31.269 A:middle L:90%
times.2. So.6. So one half of
171
00:11:31.269 --> 00:11:35.679 A:middle L:90%
3.75 times that square 23 plus 230.6. So I
172
00:11:35.679 --> 00:11:39.340 A:middle L:90%
have those together divide by two and that will be
173
00:11:39.340 --> 00:11:43.159 A:middle L:90%
D a d t again centimeter squared per minute.
174
00:11:45.039 --> 00:11:46.360 A:middle L:90%
Hope that helps.