Question

# The energy cost of horizontal locomotion as a function of the body weight of a marsupial is given by$E=22.8 w^{-11,34} .$where $w$ is the weight of the animal (in $\mathrm{kg}$ ) and $E$ is the energy expenditure (in $\mathrm{kcal} / \mathrm{kg} / \mathrm{km}$ ). Source: Wildlife Feeding and Nutrition. Suppose that the weight of a 10-kg marsupial is increasing at a rate of $0.1 \mathrm{~kg} / \mathrm{day}$. Find the rate at which the energy expenditure is changing with respect to time.

Solution

Verified
The rate of change of the energy expenditure with respect to time is determined by the derivative $dE/dt$, so first we need to compute this derivative. Taking $d/dt$ of both sides of the given equation we will get:
\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*}