Question

The rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :

• A. \(400\pi\text{ cm}^3/\text{cm}\)
• B. \(100\pi\text{ cm}^3/\text{cm}\)
• C. \(50\pi\text{ cm}^3/\text{cm}\)
• D. \(25\pi\text{ cm}^3/\text{cm}\)

Hint:

Alternatively, use the chain rule: \(\frac{dV}{dD} = \frac{dV/dr}{dD/dr} = \frac{4\pi r^2}{2} = 2\pi r^2\). Substituting \(r = 5\) directly gives \(2\pi(5)^2 = 50\pi\text{ cm}^3/\text{cm}\) in just one step!

Answer 1

The Correct Option is C

Concept:
• The volume of a sphere in terms of its radius \(r\) is \(V = \frac{4}{3}\pi r^3\).
• The diameter \(D\) is related to the radius by \(D = 2r\), which implies \(r = \frac{D}{2}\).
• The rate of change of volume with respect to diameter can be found by evaluating \(\frac{dV}{dD} = \frac{\frac{dV}{dr}}{\frac{dD}{dr}}\) or by expressing \(V\) directly in terms of \(D\).

Step 1: Express the volume in terms of diameter \(D\)
Substitute \(r = \frac{D}{2}\) into the volume formula: \[ V = \frac{4}{3}\pi \left(\frac{D}{2}\right)^3 \] \[ V = \frac{4}{3}\pi \left(\frac{D^3}{8}\right) = \frac{1}{6}\pi D^3 \]

Step 2: Differentiate the volume with respect to \(D\)
Differentiating with respect to the diameter \(D\): \[ \frac{dV}{dD} = \frac{d}{dD}\left(\frac{1}{6}\pi D^3\right) \] \[ \frac{dV}{dD} = \frac{1}{6}\pi (3D^2) = \frac{1}{2}\pi D^2 \]

Step 3: Evaluate the rate of change at the given radius
We are given that the radius is \(r = 5\text{ cm}\). The corresponding diameter is: \[ D = 2r = 2(5) = 10\text{ cm} \] Substitute \(D = 10\text{ cm}\) into the derivative formula: \[ \frac{dV}{dD} = \frac{1}{2}\pi (10)^2 = \frac{1}{2}\pi (100) = 50\pi\text{ cm}^3/\text{cm} \]