Question

A rain drop (assumed as sphere) evaporates at a rate proportional to its surface area at any instant t. The differential equation governing rate of change of radius r of the drop is

• A. $\frac{d^2r}{dt^2}+2r=0$
• B. $\frac{d^2r}{dt^2}=3r$
• C. $\frac{d^2r}{dt^2}=0$
• D. none of these

Answer 1

Answer: (C)

Since $V = \frac{4}{3}\,\pi r^{3}$
$\therefore \frac{dV}{dt} = 4\,\pi r^{2} \frac{dr}{dt}$
Given $\frac{dV}{dt} \propto 4\,\pi\,r^{2}$
$\therefore \frac{dV}{dt} = 4\,\pi \,r^{2}\,K$ (say)
$\therefore 4\,\pi r^{2} \frac{dr}{dt} = 4\pi r^{2}\,K$
$\Rightarrow \frac{dr}{dt} = K$
$\Rightarrow \frac{d^{2}r}{dt^{2}}=0$.