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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.
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.
Animal Mass (kg) Metabolic rate
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 mouse-to-elephant-curve
Kleiber modeled his data by the power function
\begin{equation*} P(m) = 73.3m^{0.74} \end{equation*}
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.