Empirical data suggest that the surface area of a 180-cm -tall human body changes at the rate of

step 1 Alright, so first we are given that our surface area of a 70 kilogram body is 1 .886277 meters s

Empirical data suggest that the surface area of a 180-cm...

Empirical data suggest that the surface area of a $180-\mathrm{cm}$ -tall human body changes at the rate of $$ S^{\prime}(W)=0.131773 W^{-0.575} $$ square meters/kilogram, where $W$ is the weight of the body in kilograms. If the surface area of a 180 -cm-tall human body weighing $70 \mathrm{~kg}$ is $1.886277 \mathrm{~m}^{2}$, what is the surface area of a human body of the same height weighing $75 \mathrm{~kg}$ ?

Empirical data suggest that the surface area of a $180-\mathrm{cm}$ -tall human body changes at the rate of
$$
S^{\prime}(W)=0.131773 W^{-0.575}
$$
square meters/kilogram, where $W$ is the weight of the body in kilograms. If the surface area of a 180 -cm-tall human body weighing $70 \mathrm{~kg}$ is $1.886277 \mathrm{~m}^{2}$, what is the surface area of a human body of the same height weighing $75 \mathrm{~kg}$ ?

Key Concepts

Initial Value Problems

Initial value problems involve determining a function that satisfies both a differential equation and an initial condition. Given S(70) in the problem, integration of S'(W) combined with the initial value permits the exact calculation of the surface area for a different weight, ensuring that the solution is uniquely determined.

Definite Integration

Definite integration is used to calculate the net change in a quantity over a specific interval. In this context, integrating S'(W) from one weight to another yields the total change in surface area between those weights, leveraging the fundamental theorem of calculus to relate the derivative and the original function.

Integration

Integration is the process of finding the accumulated total of a quantity whose rate of change is known. When given a derivative such as S'(W), integration allows us to reconstruct the original function (up to an additive constant) and compute the change in the function's value over a specified interval.

Rate of Change

Rate of change describes how a quantity changes with respect to another quantity and is typically represented by a derivative. In problems like this one, the derivative S'(W) indicates how the surface area changes as the weight changes, providing an instantaneous rate of change that can be used to calculate incremental differences.

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