Hello zcus05
Welcome to Math Help Forum! Originally Posted by
zcus05 Half of a snowball melts in an hour, how long will it take for the rest to melt? assuming that it remains spherical and that its volume decreases at a rate proportional to its surface area.
Much appreciated
I am assuming the the phrase 'half of a snowball' refers to half the volume, not that the radius has been halved (which would reduce the volume to one-eighth of its original value).
The surface area and volume formulae are:
$\displaystyle S = 4\pi r^2$
$\displaystyle V = \frac43\pi r^3$
The rate of decrease of volume is proportional to the surface area is translated as:
$\displaystyle \dfrac{dV}{dt}=-kS$
$\displaystyle \Rightarrow \dfrac{dV}{dr}\cdot\dfrac{dr}{dt}= -kS$
$\displaystyle \Rightarrow 4\pi r^2 \dfrac{dr}{dt}=-kS$
$\displaystyle \Rightarrow \dfrac{dr}{dt}=-k$
$\displaystyle \Rightarrow r = -kt+r_0$, where $\displaystyle r_0$ is the initial radius.
If the initial volume is $\displaystyle V_0$, then:
$\displaystyle V_0 = \frac43\pi r_0^3$
So when $\displaystyle V = \frac12V_0 $:
$\displaystyle \frac43\pi r^3 = \frac23 \pi r_0^3$
$\displaystyle \Rightarrow r = \dfrac{r_0}{\sqrt[3]{2}}$
So, assuming we measure time in hours, when $\displaystyle \displaystyle t = 1$:
$\displaystyle \dfrac{r_0}{\sqrt[3]{2}}=-k+r_0$
$\displaystyle \Rightarrow k = \dfrac{r_0(\sqrt[3]2-1)}{\sqrt[3]2}$
When $\displaystyle r=0$:
$\displaystyle t=\dfrac{r_0}{k}$
$\displaystyle =\dfrac{\sqrt[3]2}{\sqrt[3]2-1}$
So the time taken for the remainder to melt is:
$\displaystyle \dfrac{\sqrt[3]2}{\sqrt[3]2-1}-1=\dfrac{1}{\sqrt[3]2-1}$ hours.
If, after all, the phrase 'half the snowball' meant that the radius has been halved, then, of course the additional time taken would be one hour, since the radius is decreasing at a constant rate.
Grandad