Problem 82 - Elementary Algebra

Question

The circumference of a circle measures \(24 \pi\) inches. Find the radius of the circle.

Step-by-Step Solution

Answer

The radius of the circle is 12 inches.

Step 1: Understand the formula for circumference

The formula for the circumference of a circle is given by \( C = 2 \pi r \), where \( C \) is the circumference and \( r \) is the radius of the circle. We will use this formula to find the radius.

Step 2: Substitute the given circumference

We know the circumference \( C = 24 \pi \) inches. Substitute this value into the circumference formula: \( 24 \pi = 2 \pi r \).

Step 3: Solve for the radius

To find the radius \( r \), divide both sides of the equation \( 24 \pi = 2 \pi r \) by \( 2 \pi \). The equation becomes:\[\frac{24 \pi}{2 \pi} = r\]Simplify the left-hand side by cancelling \( \pi \), resulting in:\[\frac{24}{2} = r\]Thus, \( r = 12 \).

Step 4: Conclude the solution

We have found that the radius of the circle is \( r = 12 \) inches by following the steps above and solving the equation.

Key Concepts

Circle Radius

The radius of a circle is the distance from the center of the circle to any point on its edge. Imagine it like a spoke on a wheel, extending straight outward from the hub to the rim. Understanding the radius is crucial because it is a key component in many geometry formulas, especially those related to circles. The radius is half the length of the diameter, which is the line across the circle passing through the center. Knowing the radius helps us find the circumference, area, and even solve complex problems involving circles. It serves as the foundation for understanding circular shapes and measurements.

Circumference Formula

The circumference of a circle is the total distance around the circle. Think of it like the perimeter, or the outline that you'd trace if you walked around the edge of a circular park. The formula for calculating the circumference is critical: \[ C = 2 \pi r \]Here, \( C \) represents the circumference, \( r \) is the radius, and \( \pi \) (pi) is a special mathematical constant approximately equal to 3.14159.To solve problems involving the circumference, remember:

• Double the radius (\( 2r \)) and multiply by \(\pi\).
• Ensure you use the same measurement units for radius and circumference.

With a strong grasp of this formula, you can tackle any circle-related question effectively.

Algebraic Problem Solving

Solving algebraic problems involves carefully following steps and using mathematical operations to find unknown values. In circle-related problems, this often means manipulating equations with the circumference and radius.Let's break it down with an example from our exercise:1. **Identify Given Information:** We knew the circumference was \(24\pi\) inches.2. **Use the Relevant Formula:** We applied the formula \(C = 2\pi r\).3. **Substitute The Known Values:** Set \(24\pi = 2\pi r\).4. **Solve For The Unknown (Radius):** Divide both sides by \(2\pi\) to isolate \(r\), simplifying to \(r = 12\).These steps showcase how algebra can simplify and solve practical geometry problems, helping us find solutions logically and systematically. By understanding this process, you can solve not only for a circle's radius but also tackle a wide range of algebraic challenges.