A city’s population (in people) and land area (in square kilometers) are modeled for…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A city’s population (in people) and land area (in square kilometers) are modeled for

t\ge 0

years after January 1, 2020 by

P(t)=1.2\times10^{6}e^{0.025t}, \qquad A(t)=320+4t.

Population density is defined by

D(t)=\dfrac{P(t)}{A(t)}\quad\text{(people per km}^2\text{)}.

Answer the following questions. (a) Derive a simplified expression for

D'(t)

in terms of

t

. (b) Calculate

D'(5)

and interpret its meaning in the context of the city. (c) Find the value of

t

(to the nearest tenth of a year) at which the rate of change of density first reaches

80\text{ people km}^{-2}\text{ yr}^{-1}

. Show all work leading to your solution. (d) Describe the long-term behavior of

D'(t)

t\to\infty

. Include a brief justification based on your formula from part (a).