Let's analyze the problem based on the image you uploaded.

Part (a): Writing a Formula for the Perimeter of the Indoor Track

The indoor track consists of:

• A rectangle with length $x$ and width $2r$ (because the semicircles' radius is $r$ on both sides),
• Two semicircles with radius $r$, which combine to form a full circle.

The perimeter $P$ of the track is composed of:

1. The lengths of the two straight sides (the two sides of the rectangle), which is $2x$,
2. The perimeter of the two semicircles, which forms a complete circle. The circumference of a full circle is $2\pi r$.

Thus, the formula for the perimeter is: $P = 2x + 2\pi r$

Part (b): Solving for $x$

To solve for $x$, you can rearrange the perimeter formula: $P = 2x + 2\pi r$ Solving for $x$: $x = \frac{P - 2\pi r}{2}$

Part (c): Substituting Known Values

You are given:

• The perimeter $P = 660$ feet,
• The radius $r = 50$ feet.

Substitute these values into the formula for $x$: $x = \frac{660 - 2\pi(50)}{2}$

Let’s calculate $x$.

Calculation: