00:01
Hi there, so for this problem, we want to construct something like is shown in this figure.
00:08
So we can divide this in three segments.
00:14
Two of them are semicircles and the one at the center is a rectangle.
00:24
Now, with that said, we are told that the total distance around the inside edge of the truck is 400 meters.
00:32
So we are given the value for the perimeter of this 400 meters.
00:37
So the question is to find the maximum area for the fitness center.
00:43
So first of all, we can write the expression for the perimeter, so that will be.
00:50
If we call this size as s and s, then we will have two times x.
00:59
And then in here we're going to have the radius of this.
01:05
So that will be the perimeter of a semicircle, but since we have two semicircles, they add to one circle.
01:18
So we know that the perimeter of a circle is pi, two times pi times the radius.
01:27
Now, for the area, we can write the area of this.
01:31
The area is just x times two times the radius.
01:35
This is the area of the rectangle.
01:38
And the area for a complete circle is pi times the radius square.
01:45
So what we need to do is to, for example, from the first one, we can solve for s.
01:53
Then we will have that this is 400 minus 2 times pi times the radius.
02:01
We can simplify this by divided everything by 2...