The surface area of a human (in square meters) has been approximated by $A=0.024265 h^{0.3964} \mathrm{~m}^{0.5378}$, where $h$ 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 $72 \mathrm{~kg}$ to $73 \mathrm{~kg}$, 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 $72 \mathrm{~kg}$. Use the derivative to estimate the change.
Answer
Final Answer: The approximate change in surface area when the mass changes from 72 kg to 73 kg, while the height remains 184 cm, is \(\boxed{0.0143}\) square meters. The approximate change in surface area when the height changes from 165 cm to 166 cm, while the mass remains at 72 kg, is \(\boxed{0.0044}\) square meters.
Step 1 ：Given the surface area of a human is approximated by \(A=0.024265 h^{0.3964} m^{0.5378}\), 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 72 kg to 73 kg, while the height remains 184 cm, we first find the partial derivative of the surface area with respect to mass while keeping the height constant.
Step 3 ：The partial derivative of \(A\) with respect to \(m\) is \(A_m = 0.013049717 h^{0.3964} m^{-0.4622}\).
Step 4 ：Substituting \(h = 184\) cm and \(m = 72\) kg into \(A_m\), we get \(A_m = 0.0142861695055957\).
Step 5 ：The approximate change in surface area when the mass changes from 72 kg to 73 kg, while the height remains 184 cm, is \(\Delta A = A_m = 0.0142861695055957\) square meters.
Step 6 ：To find the approximate change in surface area when the height changes from 165 cm to 166 cm, while the mass remains at 72 kg, we find the partial derivative of the surface area with respect to height while keeping the mass constant.
Step 7 ：The partial derivative of \(A\) with respect to \(h\) is \(A_h = 0.009618646 m^{0.5378} h^{-0.6036}\).
Step 8 ：Substituting \(h = 165\) cm and \(m = 72\) kg into \(A_h\), we get \(A_h = 0.00440062183507189\).
Step 9 ：The approximate change in surface area when the height changes from 165 cm to 166 cm, while the mass remains at 72 kg, is \(\Delta A_h = A_h = 0.00440062183507189\) square meters.
Step 10 ：Final Answer: The approximate change in surface area when the mass changes from 72 kg to 73 kg, while the height remains 184 cm, is \(\boxed{0.0143}\) square meters. The approximate change in surface area when the height changes from 165 cm to 166 cm, while the mass remains at 72 kg, is \(\boxed{0.0044}\) square meters.
