What is the rate of change of the volume V of a hemisphere with respect to its radius r when r = 2?

1. Analyze the question

We are asked to find the rate of change of the volume ($V$) of a hemisphere with respect to its radius ($r$) at the specific point where $r = 2$. In calculus, the "rate of change" of a quantity with respect to another is represented by the derivative. Therefore, we need to find $\frac{dV}{dr}$ and evaluate it at $r = 2$.

2. Explain the concept

• Volume of a Sphere: The volume of a full sphere is given by $V = \frac{4}{3}\pi r^3$.
• Volume of a Hemisphere: A hemisphere is exactly half of a sphere. Therefore, the volume is: $$V = \frac{1}{2} \times \left( \frac{4}{3}\pi r^3 \right) = \frac{2}{3}\pi r^3$$
• Differentiation: To find the rate of change, we differentiate $V$ with respect to $r$ using the power rule ($\frac{d}{dx}x^n = nx^{n-1}$).

3. Show the solution

Step 1: Write the formula for the volume of a hemisphere.

Step 2: Differentiate $V$ with respect to $r$.

Using the power rule, bring down the exponent (3) and multiply:

Step 3: Evaluate the derivative at $r = 2$. Substitute $r = 2$ into the derivative formula:

4. Identify the answer

Comparing our result ($8\pi$) to the given options, it matches option A.

5. Tip

Whenever you see "rate of change" in a calculus question, immediately think of differentiation. If the question asks for the rate of change of volume with respect to radius, you are looking for $\frac{dV}{dr}$. If it asked for the rate of change with respect to time, you would use the chain rule ($\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}$).