A balloon which always remains spherical, has a variable diameter $\frac{3}{2}(2x+3)$. Determine the rate of change of volume with respect to x.

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

The correct answer is Option (3) → $\frac{27\pi}{8}(2x + 3)^2$

Radius (say r) of the spherical balloon = $\frac{1}{2}$ (diameter)

$=\frac{1}{2}.\frac{3}{2}(2x + 3)=\frac{3}{4}(2x+3)$.

Let V be the volume of the balloon, then

$V =\frac{4}{3}\pi r^3 =\frac{4}{3}\pi.\left(\frac{3}{4}(2x+3)\right)^3 =\frac{9}{16}\pi (2x+3)^3$.

∴ The rate of change of volume w.r.t. x

$=\frac{dV}{dr}=\frac{9}{16}\pi.3(2x+3)^2.2 =\frac{27}{8}\pi(2x + 3)^2$.