Int-Alg Powers and Roots

Chapter 6 Powers and Roots

mouse to elephant: Bob Elsdale/Getty Images

We next turn our attention to a large and useful family of functions, called power functions. Here is an example of a power function with fractional exponents.

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In 1932, Max Kleiber published a remarkable equation for the metabolic rate of an animal as a function of its mass. The table at right shows the mass of various animals in kilograms and their metabolic rates, in kilocalories per day. A plot of the data, resulting in the famous “mouse-to-elephant” curve, is shown in the figure.

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Animal	Mass (kg)	Metabolic rate (kcal/day)
Mouse	$0.02$	$3.4$
Rat	$0.2$	$28$
Guinea pig	$0.8$	$48$
Cat	$3.0$	$150$
Rabbit	$3.5$	$165$
Dog	$15.5$	$520$
Chimpanzee	$38$	$1110$
Sheep	$50$	$1300$
Human	$65$	$1660$
Pig	$250$	$4350$
Cow	$400$	$6080$
Polar bear	$600$	$8340$
Elephant	$3670$	$48, 800$

Kleiber modeled his data by the power function

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P (m) = 73.3 m^{0.74}

where

P

is the metabolic rate and

m

is the mass of the animal. Kleiber’s rule initiated the use of allometric equations, or power functions of mass, in physiology.

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