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
$pr−pr23=rθ,$
where $θ=p(1−123)$. 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
$43πr3[p(123−1)+1]3.$
• 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
$L3[p(123−1)+1]3.$