Chapter 8, Problem 14PT

(a)

To determine

To find: The exponential function that is used to model the population.

Answer to Problem 14PT

The exponential function that is used to model the population is $Y=a{e}^{kx}$

Explanation of Solution

Given:

The population of a city 10 years ago was 150000.

The current population is 185000

Calculation:

Consider the given condition, the expression is written as,

  $Y=a{e}^{kx}$

Here, Y is the current population, a is the population before 20 years, and k is the rate of population.

Thus, the exponential function that is used to model the population is $Y=a{e}^{kx}$

(b)

To determine

To find: The population in 25 years.

Answer to Problem 14PT

The population in 25 years is 28385550

Explanation of Solution

Given:

The population of a city 10 years ago was 150000.

The current population is 185000

Calculation:

From the part (a), calculate the value of k,

  $\begin{array}{c}Y=a{e}^{kx}\\ 185000=150000e^{10k}\\ 1.233=e^{10k}\\ \ln 1.233=10k\end{array}$

Solve further as,

  $\begin{array}{c}0.2097=10k\\ k=\frac{0.2097}{10}\\ =0.02097\end{array}$

Calculate the population after 25 years.

  $\begin{array}{c}Y=a{e}^{kx}\\ Y=150000e^{\left(0.02097\right)\left(25\right)}\\ Y=150000\left(189.237\right)\\ =28385550\end{array}$

Thus, the population in 25 years is 28385550