To find the rate of change of the volume \( V \) of a sphere with respect to its radius \( r \), we first need to understand the formula for the volume of a sphere. The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] ### Step 1: Differentiate the Volume with Respect to the 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 apply the power rule of differentiation. The power rule states that if \( f(r) = r^n \), then \( f'(r) = n \cdot r^{n-1} \). Applying this to our volume formula: \[ \frac{dV}{dr} = \frac{d}{dr} \left( \frac{4}{3} \pi r^3 \right) \] Using the constant multiple rule and the power rule: \[ \frac{dV}{dr} = \frac{4}{3} \pi \cdot 3r^{3-1} = 4\pi r^2 \] ### Step 2: Evaluate the Derivative at \( r = 1 \) Now that we have the derivative \( \frac{dV}{dr} = 4\pi r^2 \), we need to evaluate this expression at \( r = 1 \): \[ \frac{dV}{dr} \bigg|_{r=1} = 4\pi (1)^2 = 4\pi \] ### Conclusion: Correct Option Thus, the rate of change of the volume of the sphere with respect to its radius when \( r = 1 \) is: \[ \boxed{4\pi} \] This corresponds to option **B**. ### Explanation of Other Options - **Option A: 12π** - This option is incorrect because it does not follow from the differentiation process. The calculation of the derivative clearly shows that the rate of change is \( 4\pi \), not \( 12\pi \). - **Option C: 24π** - This option is also incorrect. It may arise from a misunderstanding of the differentiation process or from incorrectly applying the power rule. The correct derivative is \( 4\pi r^2 \), and substituting \( r = 1 \) gives \( 4\pi \). - **Option D: 8π** - This option is incorrect as well. It could be a result of miscalculating the derivative or misunderstanding the formula for the volume of a sphere. The correct evaluation at \( r = 1 \) yields \( 4\pi \). ### Common Pitfalls 1. **Misapplying the Power Rule**: Ensure you apply the power rule correctly when differentiating. 2. **Forgetting to Substitute**: After finding the derivative, remember to substitute the value of \( r \) to find the specific rate of change. 3. **Confusing Volume and Surface Area**: The volume formula is different from the surface area formula, which can lead to confusion. ### Revision Summary - The volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \). - The rate of change of volume with respect to radius is found by differentiating: \( \frac{dV}{dr} = 4\pi r^2 \). - Evaluating at \( r = 1 \) gives \( \frac{dV}{dr} = 4\pi \). - The correct answer is option **B: 4π**.