Question

The energy cost of horizontal locomotion as a function of the body weight of a marsupial is given by

E=22.8w11,34.E=22.8 w^{-11,34} .

where ww is the weight of the animal (in kg\mathrm{kg} ) and EE is the energy expenditure (in kcal/kg/km\mathrm{kcal} / \mathrm{kg} / \mathrm{km} ). Source: Wildlife Feeding and Nutrition. Suppose that the weight of a 10-kg marsupial is increasing at a rate of 0.1 kg/day0.1 \mathrm{~kg} / \mathrm{day}. Find the rate at which the energy expenditure is changing with respect to time.

Solution

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Answered 1 year ago
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The rate of change of the energy expenditure with respect to time is determined by the derivative dE/dtdE/dt, so first we need to compute this derivative. Taking d/dtd/dt of both sides of the given equation we will get:

ddt(E)=ddt(22.8w0.34)dEdt=22.8(0.34)w0.341ddt(w)dEdt=7.752w1.34dwdt.(1)\begin{align*} &\frac{d}{dt}(E)=\frac{d}{dt}\left(22.8w^{-0.34}\right)\\ \Longrightarrow\quad & \frac{dE}{dt}=22.8\cdot(-0.34)w^{-0.34-1}\cdot \frac{d}{dt}(w)\\ \Longrightarrow\quad & \frac{dE}{dt}=-7.752w^{-1.34}\cdot \frac{dw}{dt}.&(1) \end{align*}

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