The surface area of a human (in square meters) has been approximated by A $=0.024265 \mathrm{~h}^{0.3964} \mathrm{~m}^{0.5378}$, where $\mathrm{h}$ is the height (in $\mathrm{cm}$ ) and $\mathrm{m}$ is the mass (in $\mathrm{kg}$ )
(a) Find the approximate change in surface area if the mass changes from $69 \mathrm{~kg}$ to $70 \mathrm{~kg}$, while the height remains $184 \mathrm{~cm}$. Use the derivative to estimate the change
(b) Find the approximate change in surface area when the height changes from $165 \mathrm{~cm}$ to $166 \mathrm{~cm}$, while the mass remains at $75 \mathrm{~kg}$. Use the derivative to estimate the change
(a) Differentiate $A(m, h)$ with respect to $m$ to find $A_{m}(m, h)$
$A_{m}(m, h)=\frac{\partial}{\partial m}\left(0.024265 h^{0.3964} m^{0.5378}\right)=$
(Round to five decimal places as needed )

Step 1 ：Given the function for the surface area of a human, \(A = 0.024265h^{0.3964}m^{0.5378}\), where \(h\) is the height in cm and \(m\) is the mass in kg.

Step 2 ：To find the approximate change in surface area if the mass changes from 69 kg to 70 kg, while the height remains 184 cm, we first need to differentiate the function with respect to \(m\). This will give us the rate of change of the surface area with respect to mass.

Step 3 ：Differentiating \(A(m, h)\) with respect to \(m\) gives us \(A_{m}(m, h)\).

Step 4 ：\(A_{m}(m, h)=\frac{\partial}{\partial m}(0.024265 h^{0.3964} m^{0.5378})\)

Step 5 ：\(A_{m}(m, h)=0.013049717h^{0.3964}m^{-0.4622}\)

Step 6 ：Substituting the given values of \(h = 184\) cm and \(m = 69\) kg into the derivative function gives us \(A_{m}(m, h) = 0.0145699757467114\)

Step 7 ：The change in surface area when the mass changes from 69 kg to 70 kg, while the height remains 184 cm, is approximately the value of the derivative function at these points.

Step 8 ：\(\Delta A = A_{m}(m, h) = 0.0145699757467114\)

Step 9 ：Final Answer: The approximate change in surface area is \(\boxed{0.01457}\) square meters.