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

step 1 Right, they give us this rate of change of surface area. Formula. And they also tell us that the

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 $$S^{\prime}(W)=0.131773 W^{-0.575}$$ $\mathrm{m}^{2} / \mathrm{kg},$ where $W$ is the mass of the body in kilograms. If the surface area of a $180-\mathrm{cm}$ -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 kg?

Empirical data suggest that the surface area of a $180-\mathrm{cm}$ -tall human body changes at the rate of
$$S^{\prime}(W)=0.131773 W^{-0.575}$$
$\mathrm{m}^{2} / \mathrm{kg},$ where $W$ is the mass of the body in kilograms. If the surface area of a $180-\mathrm{cm}$ -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 kg?

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Key Concepts

Integration (Antidifferentiation)

Integration is the inverse process of differentiation and is used to recover the original function from its rate of change. Here, integrating the given derivative function with respect to mass allows us to find an explicit formula for the surface area as a function of mass.

Derivative (Rate of Change)

A derivative in calculus measures how a function changes as its input changes. In this context, the given derivative represents the rate at which the human body’s surface area changes with respect to its mass, indicating how small changes in mass affect the surface area.

Initial Value Problem

An initial value problem involves finding a particular solution to a differential equation that satisfies a specific condition given at a certain point. In this case, the known surface area at 70 kg serves as an initial condition to determine the constant of integration when reconstructing the surface area function from its derivative.

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