Chapter 3, Problem 74RE

Solution

(a)

To calculate: The function models of time.

The function is $123000{\left(0.976\right)}^t$

Given information: The population of metro Village is $123000$ and decreasing by $2.4\%$ .

Calculation:

If population decreases by $2.4\%$ each year then here are $100\%-2.4\%=9.6\%$ as many people was the year before.

Each year the population is multiplies by $0.976$, according to this model:

  $\begin{array}{c}f\left(t\right)=123000\times 0.976\times \mathrm{0.976...}\times 0.976\\ =123000\times {0.976}^t\end{array}$

(b)

To find: The year in which population will be $90000$ .

About $13\text{ years}$ .

Given information: Population of metro Ville is $123000$ and decreasing by $2.4\%$

Calculation:

From 74a, function: $123000{\left(0.976\right)}^t$ , which is return population after t years

Find the value of $t$ such that $f\left(t\right)=90000$. Substitute that in the equation:

  $90000=123000\times {0.976}^t$

Isolate exponential:

  ${0.976}^t=\frac{90}{123}$

Take logarithm of both sides and apply logarithmic properties:

  $\begin{array}{c}\ln {0.976}^t=\ln \frac{90}{123}\\ t\ln 0.976=\ln 90-\ln 123\\ t=\frac{\ln 90-\ln 123}{\ln 0.976}\\ =12.86\text{years}\end{array}$

Population should reach $90000$ in the little under $13\text{years}$ .