Frosty the Snowman

Age 16 to 18
Challenge Level

This version of the problem is almost identical to Frosty is Melting!, but here there is a rationale for why the radius decreases as it does. 

There are some possible starting points in the Getting Started section.

Here are word and pdf versions of the problem.

 

There appear to be at least three different versions of this question!

Version 1 - original STEP question from 1991

Frosty the snowman is made from two uniform spherical snowballs, of initial radii $2R$ and $3R.$ The smaller (which is his head) stands on top of the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area, the constant of proportionality being the same for both snowballs. During melting each snowball remains spherical and uniform. When Frosty is half his initial height, find the ratio of his volume to his initial volume.

If $V$ and $S$ denote his total volume and surface area respectively, find the maximum value of $\dfrac{\mathrm{d}V}{\mathrm{d}S}$ up to the moment when his head disappears.

Version 2 - Stephen Siklos' "Advanced problems in Mathematics" 2008 edition

Frosty the snowman is made from two uniform spherical snowballs, of radii $2R$ and $3R.$ The smaller (which is his head) stands on top of the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area, the constant of proportionality being the same for both snowballs. During melting, the snowballs remain spherical and uniform. When Frosty is half his initial height, show that the ratio of his volume to his initial volume is 37 : 224.

Let $V$ and $h$ denote Frosty's total volume and height at time $t$. Show that, for $2R <h \le 10R$, $$\dfrac{\mathrm{d} V}{\mathrm{d} h}=\frac{\pi} 8 (h^2 + 4R^2)$$

and derive the corresponding expression for $0 \le h < 2R$.

Sketch $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ as a function of $h$ for $4R \ge h \ge 0$.  Hence give a rough sketch of $V$ as a function of $h$.

Version 3 - Stephen Siklos' "Advanced problems in Mathematics" 2015 edition, and 2019 edition

Frosty the snowman is made from two uniform spherical snowballs, initially of radii $2R$ and $3R.$ The smaller (which is his head) stands on top of the larger. As each snowball melts, its volume decreases at a rate which is directly proportional to its surface area, the constant of proportionality being the same for each snowball. During melting, the snowballs remain spherical and uniform. When Frosty is half his initial height, show that the ratio of his volume to his initial volume is 37 : 224.

What is this ratio when Frosty is one tenth of his initial height?

 

This problem is one of a collection designed to develop students' carbon numeracy; we hope it will encourage students to think about the issues surrounding climate change.  You can find the complete collection here.