Allometry is the study of the relative size of different

Chapter , Problem 70

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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 and are related by a power law of the form = , where and are positive constants. This equation is often called the allometric equation. (a) Let be the length of a fish and be its weight. If is a typical unit of length, then 3 is a typical unit of volume. Show that the assumptions that the length of a fish is proportional to , and that its weight is proportional to its volume and therefore to 3, lead to the equation = 3. (b) Table 11.46 relates the weight of plaice32 to its length . (Plaice is a type of fish.) If = 3, what do you expect to happen if you calculate 3 for each of the entries in the table? Do this, and estimate the proportionality constant, . (c) With this choice of , plot the function = 3 and the data in Table 11.46. Is the function a reasonable model? Table 11.46 Length (cm) 33.5 34.5 35.5 36.5 37.5 38.5 Weight (gm) 332 363 391 419 455 500 Length (cm) 39.5 40.5 41.5 42.5 43.5 Weight (gm) 538 574 623 674 724