Question

A spherical naphthalene ball exposed to the atmosphere loses volume at a rate propotional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in

• A. 6 months
• B. 9 months
• C. 12 months
• D. Infinite time

Answer 1

The correct option is A 6 months

Given $(-\frac{dV}{dt})\alpha S;$ {where V, S are instantaneous volume & surface respectively}

$\Rightarrow -\frac{dV}{dt}=KS$

'K' constant of proportionality

$\because V=\frac{4}{3}\pi R^3$

S = $4\pi R^2$

$R\to radius$

$-\frac{4}{3}\pi \left(3R^2\right)\frac{dR}{dt}=K\left(4\pi R^2\right)$

$\frac{dR}{dt}=-K$

R = -Kt + C

Given at t = 0

R = 1 cm

1 = -0 + C

$\Rightarrow C=1$

$\therefore R=-Kt+1$

at t = 3 month

R = 0.5 cm

0.5 = -k(3) +1

K = $\frac{1}{6}$

R = $-\frac{t}{6}+1$

Finally, R = 0

$\Rightarrow t=6months$