Solved: Allometry is the study of the relative size of different parts of a body as a

Chapter 11, Problem 67

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Allometry is the study of the relative size of different parts of a body as a consequence of growth.31 The simplest model of allometry is one in which it is assumed that the sizes of two parts x and y are related by a power law of the form y = kxp, where k and p are positive constants. This equation is often called the allometric equation. (a) Let x be the length of a fish and y be its weight. If L is a typical unit of length, then L3 is a typical unit of volume. Show that the assumptions that the length x of a fish is proportional to L, and that its weight y is proportional to its volume and therefore to L3, lead to the equation y = kx3. (b) Table 11.47 relates the weight y of plaice32 to its length x. (Plaice is a type of fish.) If y = kx3, what do you expect to happen if you calculate x3/y for each of the entries in the table? Do this, and estimate the proportionality constant, k. (c) With this choice of k, plot the function y = kx3 and the data in Table 11.47. Is the function a reasonable model?