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 $\left(V\right)$ with radius $\left(r\right)$ is given by,
$V=\frac{4}{3}\pi r^3$
It is given that,
$d=\frac{3}{2}(2x+1)$

$\therefore r=\frac{3}{4}(2x+1)$
$\therefore V=\frac{4}{3}\pi {\left(\frac{3}{4}\right)}^3(2x+1{)}^3=\frac{9}{16}\pi (2x+1{)}^3$
Hence, the rate of change of volume with respect to $x$ is given as,
$\frac{dV}{dx}=\frac{9}{16}\pi \frac{d}{dx}(2x+1{)}^3=\frac{9}{16}\pi \times 3(2x+1{)}^2\times 2=\frac{27}{8}\pi (2x+1{)}^2$.