Question

Spherical rain drop evaporates at a rate proportional to its surface area. The differential equation corresponding to the rate of change of the radius of the rain drop if the constant of proportionality is $k >0$, is

• A. $\frac{d r}{d t}+k=0$
• B. $\frac{ dr }{ dt }- k =0$
• C. $\frac{ dr }{ dt }= kr$
• D. $\frac{ dr }{ dt }=2 kr$

Answer 1

Answer: (A)

$\frac{d v}{d t}=-k\left(4 \pi r^2\right)$
Put $v=\frac{4}{3} \pi r^3$ or $\frac{ dv }{ dt }=4 \pi r ^2 \frac{ dr }{ dt }$
Hence $\frac{ dr }{ dt }=- k$