Question

A spherical mothball has initial radius 3 cm. Due to evaporation, the radius of the ball reduces to 1 cm in 4 months. In how many months would the mothball evaporate completely if the volume is lost at a rate proportional to the surface area ?

• A. \(6\) months
• B. \(8\) months
• C. \(10\) months
• D. \(12\) months

Hint:

Volume loss proportional to surface area makes the radius shrink at a constant rate.

Answer 1

The Correct Option is A

Step 1: Model
$\dfrac{d}{dt}\left(\tfrac43\pi r^3\right) = -k(4\pi r^2)$ simplifies to $\dfrac{dr}{dt} = -k$.

Step 2: Integrate
$r = 3 - kt$. At $t = 4$, $r = 1$, so $k = 1/2$.

Step 3: Complete evaporation
$r = 0$ when $3 - t/2 = 0$, so $t = 6$ months. Option (A).

Final Answer:
Option (A). \[ \boxed{6 \text{ months}} \]