Chapter 3, Problem 20RE

Solution

To find: the exponential function that satisfies the given conditions.

  $f\left(x\right)=67,000\cdot {\left(1.0167\right)}^x$

Given information:

Initial population = 67,000 , increasing at a rate of 1.67% per year

Definition Used:

Exponential Population Model

If a population P is changing at a constant percentage rate r each year, then

  $P\left(t\right)=P_0{\left(1+r\right)}^t$   ..... (1)

Where $P_0$ is the initial population, r is expressed as a decimal, and t is time in years.

Explanation:

It is given that the initial value = 67,000, this implies that $P_0=67,000$ .

Also, population is increasing at a rate of 1.67% per day, that is, $r=1.67\%=0.0167$

Substituting this in the general form (1),

  $\begin{array}{c}P\left(t\right)=P_0{\left(1+r\right)}^t\\ =67,000{\left(1+0.0167\right)}^t\\ =67,000\cdot {\left(1.0167\right)}^t\end{array}$

This can be written as,

  $f\left(x\right)=67,000\cdot {\left(1.0167\right)}^x$

Where, x represents time.