Chapter 7.7, Problem 5GP

To determine

Write an equation to represent the population since 2010 and predict the population in 2020.

Answer to Problem 5GP

The equation to represent the population since 2010 is $y=88647\cdot {\left(0.997\right)}^t$ .

The population in 2020 is 86,023.

Explanation of Solution

Given information:

Initial population is 88,647.

Decay rate of population is 0.3%.

Calculation:

We know the exponential function equation is given by $y=a\cdot b^t$ , where a is the initial population, b is growth/decay factor of population and t is the number of years after 2010.

Also $b=1-r$ where r is decay rate.

Initial population $(a)$ = 88,647

Decay rate $(r)$ = 0.3%=0.003

Decay factor $(b)$ =1-0.003=0.997

Plugging the value of a and b in exponential function equation

$y=a\cdot b^t$

  $y=88647\cdot {\left(0.997\right)}^t$ , where t is in years.

Therefore, the equation to represent the population since 2010 is $y=88647\cdot {\left(0.997\right)}^t$ .

For year 2020 the value of t is

  $=2020-2010=10years.$

Now, plugging the value of t in $y=88647\cdot {\left(0.997\right)}^t$ equation.

$y=88647\cdot {\left(0.997\right)}^{10}$

$y\approx 86023.20$

Therefore, the population in 2020 is approximately 86,023.