Question

A spherical ball has a variable diameter $\frac{5}{2}(3x + 1)$. The rate of change of its volume w.r.t. $x$, when $x = 1$, is:

• A. $225\pi$
• B. $300\pi$
• C. $375\pi$
• D. $125\pi$

Hint:

When dealing with spherical volumes, the formula $V = \frac{4}{3}\pi r^3$ is key, and remember to apply the chain rule when differentiating with respect to a variable.

Answer 1

The Correct Option is B

The volume $V$ of a sphere is given by: \[ V = \frac{4}{3}\pi r^3 \] where $r$ is the radius of the sphere. Given that the diameter is $\frac{5}{2}(3x + 1)$, the radius is: \[ r = \frac{5}{4}(3x + 1) \] Differentiating the volume $V$ with respect to $x$, we get the rate of change of volume: \[ \frac{dV}{dx} = \frac{d}{dx}\left( \frac{4}{3} \pi \left( \frac{5}{4}(3x + 1) \right)^3 \right) \] By chain rule, this derivative can be simplified to: \[ \frac{dV}{dx} = \pi \left( \frac{5}{4} \right)^3 3(3x + 1)^2 \times \frac{d}{dx}(3x + 1) = \pi \times \frac{125}{64} \times 3(3x + 1)^2 \] Substitute $x = 1$ into the above expression: \[ \frac{dV}{dx} \bigg|_{x=1} = \pi \times \frac{125}{64} \times 3(3 \times 1 + 1)^2 = 300\pi \] Thus, the rate of change of volume when $x = 1$ is $300\pi$.