An empirical formula by E. F. Dubois relates the surface...

An empirical formula by E. F. Dubois relates the surface area $S$ of a human body (in square meters) to its weight $W$ (in kilograms) and its height $H$ (in centimeters). The formula, given by $$S=0.007184 W^{0.425} H^{0.725}$$ is used by physiologists in metabolism studies. a. Find the domain of the function $S$. b. What is the surface area of a human body that weighs $70 \mathrm{kg}$ and has a height of $178 \mathrm{cm} ?$

An empirical formula by E. F. Dubois relates the surface area $S$ of a human body (in square meters) to its weight $W$ (in kilograms) and its height $H$ (in centimeters). The formula, given by $$S=0.007184 W^{0.425} H^{0.725}$$
is used by physiologists in metabolism studies.
a. Find the domain of the function $S$.
b. What is the surface area of a human body that weighs $70 \mathrm{kg}$ and has a height of
$178 \mathrm{cm} ?$

Key Concepts

Domain of a Function

The domain of a function is the complete set of possible input values for which the function is defined. In the context of physical models and other mathematical applications, the domain must respect realistic constraints. For example, when an input represents a physical quantity such as weight or height, the domain is restricted to values that are meaningful in that real-world context—typically only positive numbers.

Empirical Formulas

Empirical formulas are mathematical relationships derived from experimental data rather than purely from theoretical considerations. They are used to predict or describe physical phenomena by fitting a formula to observed measurements, which is particularly common in fields like physiology and biology to approximate real-world behaviors such as the surface area of a human body.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously, and they are used to capture non-linear relationships in mathematical models. They indicate that the relationship between variables does not change uniformly but rather follows a specific scaling law. This is especially useful in empirical formulas where the effect of one variable on another is not simply proportional but instead follows a power-law relationship.

Substitution and Evaluation

Substitution is the process of replacing variables in an equation with specific values in order to evaluate the function. It involves performing arithmetic operations, including exponentiation with fractional exponents, to compute a numerical result. This technique is fundamental in applying mathematical models to real-world situations by obtaining specific predictions or measurements.

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