Differential Equation Problem

**eugenius** · November 4th 2008, 02:58 PM

A spherical mothball evaporates uniformly at a rate proportional to its surface area. Hence deduce a differential equation that links its radius with time. Given that the radius halves in one month, how long will the mothball last?

Thanks

**skeeter** · November 4th 2008, 04:22 PM

Originally Posted by eugenius

A spherical mothball evaporates uniformly at a rate proportional to its surface area. Hence deduce a differential equation that links its radius with time. Given that the radius halves in one month, how long will the mothball last?

Thanks

A spherical mothball evaporates uniformly at a rate proportional to its surface area.

$\frac{dV}{dt} = -kS$

since $V = \frac{4}{3}\pi r^3$

$\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt} = S \cdot \frac{dr}{dt}$

equating ...

$S \cdot \frac{dr}{dt} = -kS$

$\frac{dr}{dt} = -k$

so, the radius decreases at a constant rate ... can you take it from here?

**eugenius** · November 4th 2008, 04:26 PM

thanks, i'll see how i get on

**eugenius** · November 4th 2008, 11:50 PM

i still cant do the last bit, help please :S

**mr fantastic** · November 5th 2008, 01:18 AM

Originally Posted by eugenius

i still cant do the last bit, help please :S

$\frac{dr}{dt} = -k \Rightarrow r = -kt + C$ .

When t = 0, $r = r_0$ . Therefore $r = r_0 - kt$ .

When t = 1, $r = \frac{r_0}{2} \Rightarrow k = \frac{r_0}{2}$ .

Therefore $r = r_0 \left(1 - \frac{t}{2} \right)$ .

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