00:01
Hi there, so for this problem, we have the situation that is shown here in this figure.
00:05
We need to find the dimensions that maximize the area for this rectangle that is inscribed inside of this circle.
00:14
And we are given the radius of the circle and that radius is equal to 11.
00:20
So let's label the dimensions of the rectangle and the horizontal side of this as x and the vertical side as y.
00:32
So with that, we know that this length in here is going to correspond then to half of y.
00:44
And this distance right here is a half of x.
00:49
So with that, we know that by the pythagorean theorem, we know that 11 to the square is equal to x divided by 2, and that to the square plus y divided by two and that to the square.
01:09
So that will be that x to the square plus y to the square is equal to four times 11 to the square.
01:21
So solving for this, we will find that this is 11 to the square and this times four, which is going to give us 484.
01:35
Remember that we need to maximize the area.
01:41
So we have the area that is equal to x times y.
01:55
And then we need to maximize it.
02:00
But we need to have it in terms of only volume.
02:07
Then we can, from the previous exploration, we can solve for a for instance for y.
02:16
So y is equal to the square root of 484 minus x squared square.
02:26
And then we substitute this into the area.
02:28
So that will be that.
02:29
The area is x times the square root of 484 minus x square...