Problem 45 - Algebra for College Students

Question

The circumference of a circle is \(480 \pi\) in. (a) What is its radius? (b) What is its diameter?

Step-by-Step Solution

Answer

Radius: 240 in, Diameter: 480 in.

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. In this problem, the circumference is given as \( 480 \pi \).

Step 2 - Solve for the radius

Using the formula for circumference, set \( C = 480 \pi \): \[ 480 \pi = 2 \pi r \] To find \( r \), divide both sides by \( 2 \pi \): \[ r = \frac{480 \pi}{2 \pi} \] Simplify the expression: \[ r = 240 \] Therefore, the radius is 240 inches.

Step 3 - Solve for the diameter

The diameter \( d \) of a circle is twice the radius. Using the radius we found: \[ d = 2r \] Substitute \( r = 240 \): \[ d = 2 \times 240 = 480 \] Therefore, the diameter is 480 inches.

Key Concepts

circumference formula

The circumference of a circle is essentially the distance around its edge. Understanding the formula for circumference is crucial. The formula is given by: \[ C = 2 \pi r \] where \( C \) stands for circumference and \( r \) is the radius. Let's break it down:

• \( \pi \) (Pi) is a constant approximately equal to 3.14159.
• The factor '2' accounts for the circle being a complete round shape.
• 'r' is the radius, which is the distance from the center to any point on the circle's edge.

This formula shows that the circumference is directly proportional to the radius. This will help in finding either the radius or diameter if the circumference is known.

radius calculation

Finding the radius when you know the circumference is straightforward. Starting with the formula \( C = 2 \pi r \), we rearrange it to solve for \( r \). This is done by dividing both sides by \( 2 \pi \): \[ r = \frac{C}{2 \pi} \] For the given problem, where the circumference \( C = 480 \pi \), follow these steps:

• Substitute \( 480 \pi \) for \( C \)
• \( r = \frac{480 \pi}{2 \pi} \)
• Notice that \( \pi \) cancels out, simplifying to \( r = \frac{480}{2} \)
• Complete the division to get \( r = 240 \)

Therefore, the radius of the circle is 240 inches. This process shows how essential rearranging the circumference formula is in finding the radius.

diameter calculation

Next, let’s determine the diameter, which is an important linear measurement. The diameter is a straight line passing from one side of the circle to the other, going through the center. It is always twice the length of the radius. The relationship between the diameter \( d \) and the radius \( r \) is: \[ d = 2r \] Using the previously found radius:

• Substitute \( r = 240 \)
• \( d = 2 \times 240 = 480 \)

Thus, the diameter is 480 inches. Understanding this simple yet powerful formula helps quickly determine the size or scale of a circle when you know its radius or circumference.