Let's analyze and solve the problem step by step.

The function for the radius $r(C)$ of a circle in terms of its circumference $C$ is given by: $r(C) = \frac{C}{2\pi}$

Part (b): Find $r(24\pi)$ and interpret its meaning.

To find $r(24\pi)$, substitute $C = 24\pi$ into the function:

$r(24\pi) = \frac{24\pi}{2\pi}$

Now, simplify this expression: $r(24\pi) = \frac{24\pi}{2\pi} = \frac{24}{2} = 12$

Interpretation

The result $r(24\pi) = 12$ means that a circle with a circumference of $24\pi$ has a radius of 12 units.

The correct interpretation choice is:

• "The radius of a circle with a circumference of $24\pi$ is $r(24\pi)$."

Would you like further details on this solution, or do you have any questions?

Here are some related questions for further exploration:

1. How would the radius change if the circumference were $48\pi$ instead of $24\pi$?
2. What is the formula for the area of a circle in terms of its circumference?
3. How would you solve for $C$ in terms of $r$ if $r(C) = \frac{C}{2\pi}$?
4. What would the radius be for a circle with a circumference of $10\pi$?
5. How does the formula $r(C) = \frac{C}{2\pi}$ relate to the formula $C = 2\pi r$?

Tip:

When working with formulas, substituting values carefully and simplifying step-by-step can prevent errors and ensure clarity.