Exercise 16

There are several proposed formulas to approximate the surface area of the human body. One model uses the formula $$\begin{gathered} A(h,w) = 0.0072 h^{0.725}w^{0.425}, \end{gathered}$$ where $A$ is the surface area in square meters, $h$ is the height in centimeters, and $w$ is the weight in kilograms.

Since a person's height $h$ and weight $w$ change over time $h$ and $w$ are functions of time $t$. Let us think about what is happening to a child whose height is $60$ centimeters and weight is $9$ kilograms. Suppose, furthermore, that $h$ is increasing at an instantaneous rate of 20 centimeters per year and $w$ is increasing at an instantaneous rate of $5$ kg per year.

Determine the instantaneous rate at which the child's surface area is changing at this point in time.