00:01
This question asks us about a train.
00:06
And so i'm going to draw what's given.
00:10
I'm going to write capital t for train, and it's going in that direction.
00:17
There's a camera over here.
00:22
See, whoops, it just does not like me drawing points.
00:30
I need to figure out how to do this.
00:33
C.
00:34
Okay.
00:35
We know that this is 0 .5 kilometers.
00:41
We know that this is z kilometers and this is x kilometers.
00:49
So we're asked to express z as a function of x.
00:55
Well, it's a right triangle.
00:58
So x squared plus 0 .5 squared equals z squared.
01:02
Or z will be the square root of x squared plus 0 .5 squared.
01:15
That's it.
01:17
B.
01:25
Okay, so we want the rate of change of the distance from the camera to the train.
01:33
Okay.
01:35
So i'm going to rewrite it as z squared equals x squared plus 0 .5 squared.
01:47
Then i'm going to take the derivative of both sides with respect to t.
01:51
2 z d z over d t is 2x d x x over d t and then that's plus a constant but the derivative of a constant is zero.
02:09
So we want to know d z over d t.
02:17
First of all, the twos cancel out, and we're left with dz over dt is x over z, dx over dt.
02:36
We know dx over dt because that is 0 .8 kilometers per minute.
02:52
Also know that x is one kilometer.
02:57
So, what i'm going to do is i'm going to write 1 for x.
03:07
We don't really know what z is, but we can use the equation way up on the top here.
03:16
The square root of x squared, well, that's 1 squared, plus 0 .5 squared, times dx squared.
03:29
Over dt, excuse me, which is 0 .8.
03:37
So i'm going to put that in a calculator.
03:48
One over a square root of 1 plus 0 .5 squared times 0 .8.
04:10
So i checked with the answer in the back of the book and my answer was wrong.
04:14
And the reason is i didn't read the question carefully enough.
04:18
It says when the train is one kilometer away from the camera.
04:24
So that means when z is 1.
04:30
So let me redo this.
04:34
Erase.
04:35
We're given that z is 1.
04:41
We don't know what x is.
04:44
So i'm going to completely redo this.
04:48
Z is 1.
04:49
X if we rearrange this for x x is going to be the square root of z squared minus 0 .5 squared so we've got to put that in the numerator where x goes so z squared is 1 squared minus 0 .5 squared okay now let's see so i need to do the square root of 1 minus 0 .5 squared over 1.
05:29
Okay.
05:31
0 .693.
05:33
0 .693.
05:35
Okay, good.
05:38
So that's going to be 0 .693 kilometers per minute...