Chapter 6.4, Problem 17E

a.

To determine

To write: an exponential growth function that represents the population t years after 2000.

Answer to Problem 17E

  $y=315,000{\left(1.02\right)}^t$

Explanation of Solution

Given:

The population in the year 2000 = 315,000

Population growth rate = 2% annually

Concept used:

Exponential growth function:

  $y=a{\left(1+r\right)}^t$ ….. (1)

Here, the y is the final value, a is the initial value, and r is the rate of growth (in decimal form).

Calculation:

Substituting $a=315,000$ and $r=0.02$ in equation (1), the required exponential growth function is:

  $\begin{array}{c}y=a{\left(1+t\right)}^t\\ =315,000{\left(1+0.02\right)}^t\\ =315,000{\left(1.02\right)}^t\end{array}$

Conclusion:

So, the population t years after 2000 is given by the function $y=315,000{\left(1.02\right)}^t$ .

b.

To determine

the population in the year 2020 (round off to nearest thousand).

Answer to Problem 17E

Approximately $468,000$

Explanation of Solution

Given:

The population‘t’ years after 2000 is given by the function:

  $y=315,000{\left(1.02\right)}^t$ .

The population in the year 2020 is found by substituting $t=20$ in the above function.

  $\begin{array}{c}y\left(20\right)=315,000{\left(1.02\right)}^{20}\\ =315,000\left(1.4859\right)\\ \approx 468,073\end{array}$

Conclusion:

So, population in the year 2020 will be about 468,000.