https://file.aiquickdraw.com/mathbot/file/1727660506874ed6364bo.jpg

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.

An athletic facility is building an indoor track composed of a rectangle and two semicircles. Write a formula for the perimeter of the track, solve the formula for x, and find x when P = 660 feet and r = 50 feet.

Learn how to calculate the perimeter of an indoor track made up of a rectangle and two semicircles. This problem includes the geometric perimeter formula P = 2x + 2πr and algebraic manipulation to solve for x. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation