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

Hint:

When differentiating functions that involve powers of variables, such as \( r^2 \), apply the power rule: \( \frac{d}{dr}(r^n) = n r^{n-1} \). This simplifies the process significantly. Also, remember to substitute numerical values carefully and check cube roots or other roots to ensure the calculation is correct before substituting them into your derivative expression.

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:

\[ 6.6\pi. \]

Answer 2

The total surface area \( S \) of a hemisphere is given by: 

\[ S = 3\pi r^2. \]

Step 1: Differentiate \( S \) with respect to \( r \):

Using the power rule of differentiation: \[ \frac{dS}{dr} = \frac{d}{dr} (3\pi r^2) = 6\pi r. \]

Step 2: Evaluate \( r = \sqrt[3]{1.331} \):

We need to calculate \( r = \sqrt[3]{1.331} \). Since \( 1.1^3 = 1.331 \), we get: \[ r = 1.1. \]

Step 3: Substitute \( r = 1.1 \) into \( \frac{dS}{dr} \):

Substituting \( r = 1.1 \) into the derivative: \[ \frac{dS}{dr} = 6\pi (1.1) = 6.6\pi. \]

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

\[ 6.6\pi. \]