Single Choice

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$$\dfrac{dr}{dt}=k$$
B$$\dfrac{dr}{dt}=-k$$
Correct Answer
C$$\dfrac{d^{2}r}{dt^{2}}=k$$
D$$\dfrac{d^{2}r}{dt^{2}}=-k$$

Solution

volume $$=v=\dfrac43 \pi r^3$$ and surface area $$=S=4\pi r^2$$

$$\text{spherical rain drop evaporates at a rate proportional to its surface area}$$
$$\dfrac{dv}{dt}\propto\,S$$
$$\Rightarrow \dfrac{dv}{dt}=-kS$$ $$[\text{-k represents decrease in volume due to evaporation}]$$
$$\Rightarrow \dfrac{d}{dt}(\dfrac43\pi r^3)=-k(4\pi r^2)$$
$$\Rightarrow 4\pi r^2 \dfrac{dr}{dt}=-k\times 4\pi r^2$$
$$\therefore \dfrac{dr}{dt}=-k$$

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