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. Let $A$ be the area of the rectangular region. Label all parts of the diagram. Note that $x$ is the radius of the circular portion.  

1. Write an equation for $y$ in terms of $x$.

   Hint (a):

   circumference $=2\pi r$ (or $2\pi x$ in this case)

   Step (a):

   $200=2y+2\pi x$

2. Express $A$ as a function of $x$.

   Hint (b):

   Use the information from part (a).

   Step (b):

   $A=xy+\pi x^2$

3. What diameter will create a maximum area for the physical fitness room?

   Hint (c):

   Graph your function from part (b) on a calculator. Locate the point that represents a maximum.