Question

The rate of change (in cm2/s) of the total surface area of a hemisphere with respect to the radius r at r = 3.

• A. 66π
• B. 6.6π
• C. 3.3π
• D. 4.4π

Answer 1

Answer: (B)

6.6π

Answer 2

The total surface area $S$ of a hemisphere is given by:

$S = 3\pi r^2.$

Differentiate $S$ with respect to $r$:

$\frac{dS}{dr} = \frac{d}{dr} (3\pi r^2) = 6\pi r.$

At $r = \sqrt[3]{1.331}$, calculate $r$:

$\sqrt[3]{1.331} = 1.1 \quad (\text{since } 1.1^3 = 1.331).$

Substitute $r = 1.1$ into $\frac{dS}{dr}$:

$\frac{dS}{dr} = 6\pi (1.1) = 6.6\pi.$

Thus, the rate of change of the total surface area with respect to the radius is:

$6.6\pi.$