A frosty puddle
Problem
This problem follows on from Frosty the Snowman.
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.
Let $V$ and $h$ denote Frosty's total volume and height at time $t$.
- Show that, for $2R <h \le 10R, \qquad$ $\dfrac{\mathrm{d} V}{\mathrm{d} h}=\frac{\pi} 8 (h^2 + 4R^2)$
- 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 $h$.
This question has been adapted and changed several times since it was first published. You can see some of the various versions via the Teachers Resources tab.
Adapted from STEP Mathematics I, 1991, Q2. Question reproduced by kind permission of Cambridge Assessment Group Archives. The question remains Copyright University of Cambridge Local Examinations Syndicate ("UCLES"), All rights reserved.
Getting Started
This problem follows on from Frosty the Snowman so the results from this problem will be useful here.
Why do you think the first request asks you to consider the range $2R<h \le 10R$?
In the range $2R < h \le 10R$, can you express $V$ in terms of $h$?
Can you differentiate your expression for $V$ to find $\dfrac{\mathrm{d} V}{\mathrm{d} h}$?
What can you say about the situation when $h<2R$? Can you express $h$ in terms of $t$? Can you use this to express $V$ in terms of $h$ and differentiate?
The sketch of $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ will be in two parts. You will need to show clearly what happens at the $h=2R$ boundary.
How are the graphs of $V$ and $\dfrac{\mathrm{d} V}{\mathrm{d} h}$ related?
Student Solutions
Well done to Moosa from St. Olave's Grammar School in the UK who sent this full solution. Click here to see a larger version.
Teachers' Resources
This problem follows on from Frosty the Snowman.
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
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
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.