Question

A balloon, which always remains spherical has a variables radius. Find the rate at which its volume is increasing with the radius when the later is $10\text{cm}$.

Answer 1

Let $x$ be the radius of the spherical balloon Volume of balloon is,

$V=\frac{4}{3}\pi x^3$

Now,

$\frac{dV}{dx}=\frac{d}{dx}\left(\frac{4}{3}\pi x^3\right)$

$=\frac{4}{3}\pi .3x^2$

$=4\pi x^2$

When $x=10$

$\frac{dV}{dx}=4\pi (10{)}^2=400\pi ‘{\text{cm}}^3/\text{cm}$

Therefore, the volume is increasing at the rate of $400\pi {\text{cm}}^3/\text{cm}$