Question

Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If k is a positive constant then differential equation involving the rate of change of the radius of the rain drop is

• A. $\frac{dr}{dt}=k$
• B. $\frac{dr}{dt}=-k$
• C. $\frac{d^{2}r}{d{t}^2}=k$
• D. $\frac{d^{2}r}{d{t}^2}=-k$

Answer 1

Answer: (B)

$\frac{dr}{dt}=-k$

volume $=v=\frac{4}{3}\pi r^3$ and surface area $=S=4\pi r^2$
$\text{spherical rain drop evaporates at a rate proportional to its surface area}$
$\frac{dv}{dt}\propto S$
$\Rightarrow \frac{dv}{dt}=-kS$ $\left[\text{-k represents decrease in volume due to evaporation}\right]$
$\Rightarrow \frac{d}{dt}\left(\frac{4}{3}\pi r^3\right)=-k\left(4\pi r^2\right)$
$\Rightarrow 4\pi r^2\frac{dr}{dt}=-k\times 4\pi r^2$
$\therefore \frac{dr}{dt}=-k$