# Intermediate Algebra: Functions and Graphs

## Chapter6Powers and Roots

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(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
\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.