Question

A spherical ball of salt is dossolving in water in such a manner that the rate of decrease of the volume at any instant is propotional to the surface. Prove that the readius is decreasing at a constant rate.

Answer 1

We have, rate of decrease of the volume of spherical ball of salt at any instant is $\propto$ surface Let the radius of the sperical ball of the salt be r.
$\therefore$ Volume of the ball (v) $=\frac{4}{3}\pi r^3$
and surface area $\left(S\right)=4\pi r^2$
$\because \frac{dV}{dt}\propto S\Rightarrow \frac{d}{dt}\left(\frac{4}{3}\pi r^3\right)\propto 4\pi r^2\Rightarrow \frac{4}{3}\pi .3r^2.\frac{dr}{dt}\propto 4\pi r^2\frac{dr}{dt}\propto \frac{4\pi r^2}{4\pi r^2}\Rightarrow \frac{dr}{dt}k.1\Rightarrow \frac{dr}{dt}=k$