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 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 ?
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Volume loss proportional to surface area makes the radius shrink at a constant rate.
Updated On: Oct 1, 2026
- \(6\) months
- \(8\) months
- \(10\) months
- \(12\) months
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Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
Let the radius be \(r\). Volume is \(V = \frac43\pi r^3\) and surface area is \(S = 4\pi r^2\). The volume decreases at a rate proportional to the surface area: \(\dfrac{dV}{dt} = -kS\).
Step 2: Form the equation
\[ 4\pi r^2\frac{dr}{dt} = -k\,4\pi r^2 \Rightarrow \frac{dr}{dt} = -k \]
So the radius shrinks at a constant rate \(k\) per month.
Step 3: Use the data
The radius falls from 3 cm to 1 cm in 4 months, a drop of 2 cm. So \(k = \dfrac24 = 0.5\) cm per month.
The remaining 1 cm takes \(\dfrac{1}{0.5} = 2\) more months. So the total time for the radius to go from 3 cm to 0 is \(4 + 2 = 6\) months.
This is option (A). The mothball therefore evaporates completely in 6 months.
Final Answer:
The mothball evaporates completely in 6 months, option (A). \[ \boxed{6 \text{ months}} \]
Let the radius be \(r\). Volume is \(V = \frac43\pi r^3\) and surface area is \(S = 4\pi r^2\). The volume decreases at a rate proportional to the surface area: \(\dfrac{dV}{dt} = -kS\).
Step 2: Form the equation
\[ 4\pi r^2\frac{dr}{dt} = -k\,4\pi r^2 \Rightarrow \frac{dr}{dt} = -k \]
So the radius shrinks at a constant rate \(k\) per month.
Step 3: Use the data
The radius falls from 3 cm to 1 cm in 4 months, a drop of 2 cm. So \(k = \dfrac24 = 0.5\) cm per month.
The remaining 1 cm takes \(\dfrac{1}{0.5} = 2\) more months. So the total time for the radius to go from 3 cm to 0 is \(4 + 2 = 6\) months.
This is option (A). The mothball therefore evaporates completely in 6 months.
Final Answer:
The mothball evaporates completely in 6 months, option (A). \[ \boxed{6 \text{ months}} \]
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