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 SAY ratio tend to retain heat whereas animals with a high SAY ratio (such as children and hummingbirds) lose heat relatively quickly. a. What is the SAY ratio of a cube with side lengths R?
b. What is the SAY ratio of a ball with radius R?
c. For a fixed constant h>0, consider a ball of radius 3√h / 4 and an ellipsoid with a long axis of length 2 3√h ​and short axes that are each half the length of the long axis. Show that the ball and ellipsoid have the same volume. (Hint: The volume of an ellipsoid is 4/3π ABC , where the axes have lengths 2A, 2B, and 2C).
d. Create a table to compare SAY ratios of the balls and ellipsoids of equal volume described in part (c) using values of h=1.1, 5, 10, and 20.
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?