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.

Solution to Question 1:

  • The volume of the larger sphere is 4/3π(pr)3, so the volume remaining is
    43π(p3r3/2);i.e.,43π(pr/23)3,
    so the radius has shrunk by an amount
    prpr23=rθ,
    where θ=p(1123). This reduction also applies to the smaller snow sphere, so its new radius is r = r(1 − θ).
    Hence, the volume of the smaller sphere is now
    43πr3[p(1231)+1]3.
  • For p = 2, 3, and 4, the proportions remaining are approximately 0.2027, 0.0535, and 0.0053, respectively.

  • The factor (1 − θ) = 0 when p ≈ 4.847.

Solution to Question 2: Following the same argument as in Question 1, it is readily shown that the side length of the smaller cube is now L(1 − θ), and its volume is
L3[p(1231)+1]3.