Question

A balloon, which always remains spherical, has a variable diameter $\frac{3}{2}(2x+1)$ Find the rate of change of its volume with respect to $x$

Answer 1

The volume of a sphere $(V)$ with radius $(r)$ is given by,
$v=\frac{4}{3} πr^3$
It is given that:
$=\frac{3}{2}(2x+1)$
Diameter
$\implies r=\frac{3}{4}(2x+1)$
$∴ v=\frac{4}{3} π(\frac{3}{4})^3(2x+1)^3=\frac{9}{16}π(2x+1)^3$
Hence, the rate of change of volume with respect to $x$ is as
$\frac{dv}{dx}=\frac{9}{16}π\frac{d}{dx}(2x+1)^3=\frac{9}{16}πx^3(2x+1)^2=\frac{27}{8}π(2x+1)^2.$