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:

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
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