An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter running track.

a) Draw a figure that visually represents the problem. Let x and y represent the length and width of the rectangular region.
b) Determine the radius of the semicircular ends of the track.
c) Use the result of part b to write an equation in terms of x and y, for the distance traveled in one lap around the track. Solve for x.
d) Use the result of part c to write the area A of the rectangular region as a function of x.

length of rectangle -- x

width of rectangle --- y

b) radius of semicircle is y/2

c) distance = 2 lengths + 2 halfcircles
= 2x +2(1/2) π(y/2)^2
= 2x + π(y^2)/4

2x + (1/4)πy^2 = 200
times 4
8x + πy^2 = 800
x = (800 - πy^2)/8

d) -- poorly worded question.
Since they had you solve for x in c) they should have asked for the area in terms of y , not x

check your typing.

BTW, I would have defined the radius of the semicircle as r
then the width of the rectangle would be 2r, and we can avoid some of these nasty fractions.

a) To visually represent the problem, we can draw a rectangular region with semicircles on each end. The length of the rectangular region is represented by x, and the width is represented by y. The track runs along the perimeter of the indoor fitness room. Here is a diagram:

```
____________________________________
| y | |
| | |
| | x |
| | |
|____________________|______________|
semicircle semicircle
```

b) To determine the radius of the semicircular ends of the track, we need to consider that the perimeter of the room is a running track. The formula for the perimeter of a rectangle is P = 2 * (length + width), and the formula for the circumference of a circle is C = 2 * π * r, where r is the radius.

In this case, since there are two semicircles, the total length of the track is equal to the perimeter of the rectangle plus the circumference of the two semicircles. Therefore, we can set up the equation:

200 = 2 * (x + y) + 2 * (π * r)

Since we want to find the radius of the semicircle, we can rearrange the equation to solve for r:

200 - 2 * (x + y) = 2 * π * r

Divide both sides by 2 * π:

(200 - 2 * (x + y)) / (2 * π) = r

So, (200 - 2 * (x + y)) / (2 * π) gives us the radius of the semicircular ends of the track.

c) Now, we can write an equation in terms of x and y for the distance traveled in one lap around the track. The distance traveled is equal to the distance around the rectangular part of the room plus the distance around the two semicircular ends.

Distance = 2 * (x + y) + 2 * (π * r)

Using the result from part b, we can substitute the expression for r:

Distance = 2 * (x + y) + 2 * (π * [(200 - 2 * (x + y)) / (2 * π)])

Simplifying, we get:

Distance = 2 * (x + y) + 200 - 2 * (x + y)

The term 2 * (x + y) cancels out, leaving:

Distance = 200 meters

Therefore, the distance traveled in one lap around the track is always 200 meters, regardless of the values of x and y.

d) Finally, we can write the area A of the rectangular region as a function of x. The area of a rectangle is given by A = length * width.

In this case, the length is x, and the width is y. Therefore:

A = x * y

Hence, the area A of the rectangular region is simply the product of x and y.