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 y³ - 8000x² = 0.

(a) Use implicit differentiation to find a formula for dy/dx, the rate of change of library circulation with respect to population size.

(b) Find the value of dy/dx when x = 27 thousand persons and y = 180 thousand books per year.

(c) Assume that x and y are both differentiable functions of time t, and use the method of related rates to find a formula for dy/dt , the time rate of change of library circulation.

(d) Compute dy/dt when x = 27, y = 180, and the population is rising at the rate of 1.8 thousand persons per year

Either use part (c) or use part (b) and the chain rule.

Transcribed Image Text:

dx dt = 1.8