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