Chapter 3, Problem 48RE

Use of Books at a Library A town library estimates that, when the population is $x$ thousand persons, approximately $y$ thousand books will be checked out of the library during $1$ year, where $x$ and $y$ are related by the equation $y^3-8000x^2=0$.

Use implicit differentiation to find a formula for $\frac{dy}{dx}$, the rate of change of library circulation with respect to population size.

Find the value of $\frac{dy}{dx}$ when $x=27$ thousand persons and $y=180$ thousand books per year.

Assume that $x$ and $y$ are both differentiable functions of time $t$, and use the method of related rates to find a formula for $\frac{dy}{dt}$, the time rate of change of library circulation.

Compute $\frac{dy}{dt}$ when $x=27,y=180$, and the population is rising at the rate of $1.8$ thousand persons per year $\left(\frac{dx}{dt}=1.8\right)$. Either use part (c) or use part (b) and chain rule.