0:00
All right.
00:01
So in this problem, we want to find the dimensions of a rectangle that will give us the maximum area that we can inscribe in the circle.
00:09
I'm kind of confused by these little question marks here, but i'm assuming the radius is supposed to, it's supposed to say r equals 15.
00:17
So i'm assuming the radius is 15.
00:19
So here's how you do it.
00:21
Here's the circle with the radius of 15.
00:25
Here's the rectangle.
00:28
I don't know what it really looks like.
00:29
But here's a rectangle.
00:30
Rectangle inscribed in the circle.
00:33
As best i can draw it.
00:34
Sorry.
00:35
It's supposed to be all these corners, all the corners of the rectangle are supposed to hit the circle.
00:40
My artistic skills, especially with a stylus on a touchscreen, not the best.
00:47
Okay.
00:48
Now, the radius is this right here.
00:51
So here's the center of the circle.
00:53
If i go to the diagonal, along the diagonal, this is 15.
00:59
All right.
01:00
We want to find the dimensions.
01:02
So i'm going to call this side because it's going left and right.
01:06
I'm just going to call it x.
01:07
I'm going to call this side going up and down y, so i know that the area is equal to x times y.
01:15
All right, great, but that's two variables.
01:18
We don't want two variables.
01:19
We only want one.
01:20
Okay, let's find it.
01:22
If i can kind of finish off this little triangle here, draw this way and draw up.
01:30
This little green triangle, this side here is going to be half of the bottom of the rectangle.
01:38
This little side here is going to be y over 2 because it's half of that side of the rectangle.
01:45
So using the pythagorean theorem, 15 squared is going to be x squared over 4.
01:52
I got to square the 2 also plus y squared over 4.
01:56
15 squared is 225.
02:00
If you multiply by 4, you get 900.
02:03
So, 900 is equal to x squared plus y squared.
02:07
And this is useful because now we can solve for y.
02:11
Why do we always solve for y? i have no idea.
02:14
But 900 minus x squared equals y squared.
02:19
So y is going to equal the square root of 900 minus x squared.
02:27
Substitute that in.
02:28
So our area formula now with just x, i'll even put a little x there for you, area with respect to just x is the x, but now i'm going to put in that green part, the square root of 900 minus x squared.
02:46
There's our area where x is the only variable.
02:49
Well, if we want to maximize it, we take the derivative and figure out where the derivative is equal to zero.
02:54
All right, so a prime of x product rule.
02:59
I didn't leave myself a lot of room.
03:01
Let me move it somewhere else.
03:05
I have some room.
03:06
I'll do it down here.
03:07
A prime of x, we do the product rule.
03:10
The derivative of the x is just one times the square root part, 900 minus x squared, plus now we just leave the x and we've got to take the square root of the green piece, the green part...