Let's go through the steps to solve this problem:

a. Formula for the Perimeter $P$ of the Indoor Track:

The track consists of a rectangle in the middle and two semicircles on either end.

• The length of the rectangle is $x$.
• The width of the rectangle is $2r$ (since $r$ is the radius of each semicircle, so two radii make the full width).
• The perimeter consists of the two long sides of the rectangle, plus the circumferences of the two semicircles (which together form a full circle).

The formula for the perimeter is:

$P = 2x + 2\pi r$

Explanation:

• $2x$ accounts for the two long sides of the rectangle.
• $2\pi r$ is the circumference of the full circle formed by the two semicircles.

b. Solve the Formula for $x$:

To isolate $x$, rearrange the formula for the perimeter:

$P = 2x + 2\pi r$

Solving for $x$:

$2x = P - 2\pi r$ $x = \frac{P - 2\pi r}{2}$

c. Given $P = 660$ feet and $r = 50$ feet, find $x$:

Substitute the values of $P = 660$ and $r = 50$ into the equation:

$x = \frac{660 - 2\pi \times 50}{2}$

Now, let's calculate this. I'll work out the result.