To find the rate of change of the volume \( V \) of a hemisphere with respect to its radius \( r \) when \( r = 2 \), we first need to understand the formula for the volume of a hemisphere. ### Step 1: Volume of a Hemisphere The formula for the volume \( V \) of a hemisphere is given by: \[ V = \frac{2}{3} \pi r^3 \] ### Step 2: Differentiate the Volume with Respect to Radius To find the rate of change of the volume with respect to the radius, we need to differentiate \( V \) with respect to \( r \). This means we will find \( \frac{dV}{dr} \). Using the power rule of differentiation, we differentiate \( V \): \[ \frac{dV}{dr} = \frac{d}{dr} \left( \frac{2}{3} \pi r^3 \right) \] Applying the power rule, where \( \frac{d}{dr}(r^n) = n r^{n-1} \): \[ \frac{dV}{dr} = \frac{2}{3} \pi \cdot 3 r^{3-1} = 2 \pi r^2 \] ### Step 3: Evaluate the Derivative at \( r = 2 \) Now, we need to evaluate \( \frac{dV}{dr} \) at \( r = 2 \): \[ \frac{dV}{dr} \bigg|_{r=2} = 2 \pi (2^2) = 2 \pi \cdot 4 = 8\pi \] ### Conclusion: Correct Option Thus, the rate of change of the volume of the hemisphere with respect to its radius when \( r = 2 \) is: \[ \frac{dV}{dr} = 8\pi \] The correct option is **A. 8π**. ### Step 4: Explanation of Other Options - **Option B: 16π** - This option is incorrect because it suggests that the rate of change is double the correct value. This could arise from a misunderstanding of the differentiation process or miscalculating the evaluation at \( r = 2 \). - **Option C: 2π** - This option is too low. It may come from incorrectly applying the power rule or not fully evaluating the expression \( 2 \pi r^2 \) at \( r = 2 \). - **Option D: 4π** - This option is also incorrect. It might result from miscalculating the square of the radius or misunderstanding the formula for the volume of a hemisphere. ### Common Pitfalls - **Misapplying the Power Rule**: Ensure you apply the power rule correctly when differentiating. - **Forgetting to Evaluate**: After finding the derivative, remember to substitute the value of \( r \) to find the specific rate of change. - **Confusing Volume Formulas**: Make sure you are using the correct formula for the volume of a hemisphere, as it differs from that of a full sphere. ### Revision Summary - The volume of a hemisphere is given by \( V = \frac{2}{3} \pi r^3 \). - The rate of change of volume with respect to radius is found by differentiating: \( \frac{dV}{dr} = 2 \pi r^2 \). - Evaluate the derivative at the given radius to find the specific rate of change. - The correct answer for \( r = 2 \) is \( 8\pi \).