Question

A sperical ball of salt is dissolving in water in such a manner that the rate of decrease of its volume at any instant is equal four times its surface area. The rate of decrease in its radius (in units/sec) will be

• A. $1$
• B. $4$
• C. $2$
• D. $\frac{4}{3}$

Answer 1

The correct option is B $4$

Let $V$ be the volume
So, $-\frac{dV}{dt}=4\left(\text{surface area}\right)$
$\Rightarrow -\frac{d}{dt}\left(\frac{4}{3}\pi r^3\right)=4\left(4\pi r^2\right)$
$\Rightarrow -\frac{4}{3}\pi .3r^2\frac{dr}{dt}=16\pi r^2$
$\Rightarrow \frac{dr}{dt}=-4\text{units/sec}$
$\therefore$ Rate of decrease in its radius is $4\text{units/sec}$