Frosty the Snowman

Age 16 to 18
Challenge Level

You might find the following useful!

  • Surface area of a sphere:  $A=4 \pi r^2$
  • Volume of a sphere: $V=\frac 4 3 \pi r^3$

Can you write down a differential equation connecting $V$ and $A$, using what you have been told about the rate at which the volume decreases?

We are told that for each snowball the volume decreases at a rate which is proportional to the surface area, so we have: $$\dfrac{\mathrm{d} V}{\mathrm{d} t}=-kA \quad \text{where} \quad k>0$$

Can you rewrite the differential equation in terms of $r$?

Substituting for $V$ and $A$ gives: $$ \quad \; \; \; \dfrac{\mathrm{d}}{\mathrm{d} t}(\frac 4 3 \pi r^3)=-k\times 4 \pi r^2$$ $$ \; \;  \implies 4 \pi r^2 \dfrac{\mathrm{d} r}{\mathrm{d} t} =-k\times 4 \pi r^2 $$ $$ \implies \dfrac{\mathrm{d} r}{\mathrm{d} t} =-k$$

It might be helpful to use notation like $r_1=$ radius of head, $r_2=$ radius of body. 

Can you solve the differential equation to find $r_1$ and $r_2$ in terms of $t$?

Integrating with respect to $t$ gives: $$r=-kt + c$$ Using the initial values of $r_1$ and $r_2$ gives: $$r_1=-kt+2R$$ $$r_2=-kt+3R$$

Can you write the height of Frosty ($h$) as a function of $r_1$ and $r_2$? (it might help to draw a sketch of Frosty!)  Can you write $h$ in terms of $t$?

If you draw a picture of two circles, one on top of the other, you should find that $h=2r_1+2r_2$. Substituting the expressions for $r_1$ and $r_2$ gives $h=2(-kt+2R)+2(-kt+3R) = -4kt+10R$

What is the initial value of $h$?  Can you find an expression for $t$ when the height is half the initial height?  

Can you use this value of $t$ to find out the radii of the two snowballs at this time?  Use this to find the volumes of each one.

What is the total volume of Frosty at the start?  What is the total volume when the height is half the initial height?  Can you find a simplify a ratio between these two total volumes?

At what point does Frosty's head disappear?

In terms of $R$, how tall is Frosty when his height is one tenth of the initial height?  What does that tell you?