Chapter 6.6, Problem 39E

Surface-area-to-volume ratio (SAV) In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.

1. a. What is the SAV ratio of a cube with side lengths a?
2. b. What is the SAV ratio of a ball with radius a?
3. c. Use the result of Exercise 34 to find the SAV ratio of an ellipsoid whose long axis has length $2a\sqrt[3]{4}$, for a ≥ 1, and whose other two axes have half the length of the long axis. (This scaling is used so that, for a given value of a, the volumes of the ellipsoid and the ball of radius a are equal.) The volume of a general ellipsoid is $V=\frac{4\pi }{3}ABC$, where the axes have lengths 2A, 2B, and 2C.
4. d. Graph the SAV ratio of the ball of radius a ≥ 1 as a function of a (part (b)) and graph the SAV ratio of the ellipsoid described in (part (c)) on the same set of axes. Which object has the smaller SAV ratio?
5. e. Among all ellipsoids of a fixed volume, which one would you choose for the shape of an animal if the goal is to minimize heat loss?