Question

The rate of change of surface area of a sphere with respect to its radius \( r \), when \( r = 4 \, {cm} \), is:

• A. \( 64\pi \, {cm}^2/{cm} \)
• B. \( 48\pi \, {cm}^2/{cm} \)
• C. \( 32\pi \, {cm}^2/{cm} \)
• D. \( 16\pi \, {cm}^2/{cm} \)

Hint:

For problems involving rates of change, first find the formula for the quantity of interest, differentiate with respect to the variable, and substitute the given value.

Answer 1

The Correct Option is C

Step 1: Define the surface area of a sphere. The formula for the surface area \( S \) of a sphere is: \[ S = 4\pi r^2. \]
Step 2: Calculate the derivative of \( S \) with respect to \( r \). This represents the rate of change of surface area with respect to the radius:
\[ \frac{dS}{dr} = \frac{d}{dr} (4\pi r^2) = 8\pi r. \]
Step 3: Substitute the given radius, \( r = 4 \, {cm} \).
\[ \frac{dS}{dr} = 8\pi (4) = 32\pi \, {cm}^2/{cm}. \]
Conclusion: The rate of change of the surface area is \( 32\pi \, {cm}^2/{cm} \).