Question

The rate of change of the area of a circle with respect to its radius \( r \) (in cm\(^2\)/cm) at \( r = 6 \) cm is:

• A. \( 10\pi \)
• B. \( 12\pi \)
• C. \( 8\pi \)
• D. \( 11\pi \)

Hint:

For problems involving rate of change: - Write the formula for the quantity (e.g., area of a circle). - Differentiate with respect to the given variable. - Plug in the specific value to find the rate at that point.

Answer 1

The Correct Option is B

Step 1: The area \( A \) of a circle in terms of radius \( r \) is: \[ A = \pi r^2 \] Step 2: To find the rate of change of area with respect to radius, differentiate \( A \) with respect to \( r \): \[ \frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r \] Step 3: Substitute \( r = 6 \) cm: \[ \frac{dA}{dr}\Big|_{r=6} = 2\pi \cdot 6 = 12\pi \text{ cm}^2/\text{cm} \] Final Answer: \( \boxed{12\pi} \)