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

The Correct Option is B

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:

\[ 6.6\pi. \]