Problem 46 - College Algebra with Corequisite Support

Question

The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2 \pi r\). Express the radius of a circle as a function of its circumference. Call this function \(r(C)\). Find \(r(36 \pi)\) and interpret its meaning.

Step-by-Step Solution

Answer

The radius of a circle with circumference \(36\pi\) is 18 units.

Step 1: Express the Radius as a Function of Circumference

To express the radius of a circle as a function of its circumference, start with the original formula for the circumference: \[ C = 2 \pi r \]Now, solve for \( r \) by isolating it on one side of the equation. Divide both sides by \( 2 \pi \) to obtain:\[ r = \frac{C}{2 \pi} \] Therefore, the function that expresses the radius \( r \) in terms of the circumference \( C \) is:\[ r(C) = \frac{C}{2 \pi} \]

Step 2: Calculate the Radius for a Given Circumference

Use the function \( r(C) = \frac{C}{2 \pi} \) to find the radius when the circumference \( C = 36 \pi \). Substitute \( C = 36 \pi \) into the function:\[ r(36 \pi) = \frac{36 \pi}{2 \pi} \] Simplify the fraction:\[ r(36 \pi) = \frac{36}{2} \] \[ r(36 \pi) = 18 \]

Step 3: Interpret the Result

The result \( r(36 \pi) = 18 \) means that the radius of a circle with a circumference of \( 36 \pi \) units is 18 units. This tells us how far from the center of the circle the boundary is when its circumference is \( 36 \pi \).

Key Concepts

Circumference of a Circle

The circumference of a circle is the distance around the circle's boundary or the total length of its outer edge. It is similar to the perimeter of a polygon.
The formula to calculate the circumference (C) is given by:

• C = 2 \pi r

Here, \( r \) is the radius of the circle, which is the distance from the center of the circle to any point on its edge.
The constant \( \pi \) (pi) is approximately 3.14159, a mathematical constant that represents the ratio of a circle's circumference to its diameter.
Knowing this formula is crucial because it allows us to find the circumference if the radius is known, and vice versa.

Radius as a Function of Circumference

The task of expressing the radius as a function of the circumference involves rearranging the original formula for circumference.
Given the formula for circumference \( C = 2 \pi r \), the goal is to solve for \( r \) in terms of \( C \).
Here's how you do it:

• Start with the equation: \( C = 2 \pi r \)
• To isolate \( r \), divide both sides by \( 2 \pi \):
• \( r = \frac{C}{2 \pi} \)

This result means that the radius \( r \) can be viewed as a function of the circumference \( C \).
So, we can express it as \( r(C) = \frac{C}{2 \pi} \).
This function allows us to determine the radius from any given circumference. For example, if a circle has a circumference of \( 36 \pi \), we substitute \( 36 \pi \) into our function: \( r(36 \pi) = \frac{36 \pi}{2 \pi} \), resulting in a radius of 18.

Solving Equations

Solving equations involves finding values that make the equation true. In the context of circles, we often solve equations to find the radius, diameter, or other properties based on known measurements like the circumference.
Here's the process applied to finding the radius from the circumference:

• Given: \( C = 36 \pi \)
• Use the function \( r(C) = \frac{C}{2 \pi} \)
• Substitute \( C \) with \( 36 \pi \):
• \( r(36 \pi) = \frac{36 \pi}{2 \pi} \)

Simplifying involves canceling \( \pi \) from numerator and denominator, yielding \( 36/2 = 18 \).
This means the radius is 18 when the circumference is \( 36 \pi \).
Solving such equations enhances understanding of how the properties of a circle are interrelated and how one property determines others. It's fundamental in geometry and useful in various real-world applications, from engineering to everyday problem-solving.