In standard calculus textbooks—usually as a problem on related rates—it is shown that for a snowball melting at a rate proportional to its surface area, its radius decreases at a constant rate regardless of its initial size. A further assumption (usually unstated) is that the snowball remains spherical as it melts.

Question 1:

  • Consider two such “snow spheres” of radius r(t) and pr(t), where p > 1 and t is time. When half of the larger sphere has melted, what proportion of the smaller one remains?

  • Evaluate the answer to part (i) for p = 2, 3, and 4.

  • For what value of p does the smaller sphere first melt?

Question 2: Show that the same volume reduction factor applies for cubes of side L(t) and pL(t).

(Assume that the cubes are suspended from a corner so that all faces are exposed to warm air.)

Fermi Questions are brief questions with answers and back-of-the-envelope estimation techniques. To submit ideas, please email John Adam (