Question

If a spherical balloon has a variable diameter $3x+(9/2)$, then the rate of change of its volume with respect to $x$ is

• A. $27\pi {(2x+3)}^2$
• B. $\frac{27\pi }{16}{(2x+3)}^2$
• C. $\frac{27\pi }{8}{(2x+3)}^2$
• D. None of these

Answer 1

The correct option is C

$\frac{27\pi }{8}{(2x+3)}^2$

Find the rate of change in volume:

Given

The diameter of the balloon $d=3x+(9/2)$

The radius of the balloon $r=\frac{1}{2}\left(3x+\frac{9}{2}\right)=\frac{3x}{2}+\frac{9}{4}$

$\Rightarrow \frac{dr}{dx}=\frac{3}{2}$

We know that volume of the spherical balloon is $\frac{4}{3}\pi {\left(r\right)}^3$

Rate of change in volume $dv/dx$

$$\begin{gathered} =\frac{4}{3}\pi \times 3r^2\left(\frac{dr}{dx}\right) \\ =\frac{4}{3}\pi \times 3{\left(\frac{3x}{2}+\frac{9}{4}\right)}^2\left(\frac{3}{2}\right) \\ =4\pi {\left(\frac{3}{4}\right)}^2{(2x+3)}^2\left(\frac{3}{2}\right) \\ =\left(\frac{27\pi }{8}\right){(2x+3)}^2 \end{gathered}$$

Hence, the correct option is C.