Question

9. The population of a town is 2500 and is decreasing at a rate of 35% per year. Write an exponential function to find the population of the town after 5 years y - lal (16) + (el) Please enter the appropriate values for a, b, c and d 2500 0085 The population after 5 years would be 2093 Round your answer to the nearest whole number (you can not have part of a living being)

Answer 1

For this, you can first find the annual decrease factor, like this

*You can express the percentage like this

`35\text{ \%}=(35)/(100)=0.35`

*Then, for the annual decrease factor, you have

`\begin{gathered} 2500\cdot0.35=875 \\ 2500-875=1625\Rightarrow\text{ Population after one year} \\ \text{ So, the annual decrease factor is} \\ (2500)/(1625)=(20)/(13)\approx1.54 \end{gathered}`

Now, the exponential decay is modeled by the equation

`\begin{gathered} y=ab^(-x) \\ \text{ Where} \\ a=\text{ initial population} \\ b=\text{ annual decrease factor} \\ x=\text{time in years} \end{gathered}`

So, then the exponential function to find the population of the city after x years is

`y=(2500)((20)/(13))^(-x)`

Then, the population after 5 years will be

`\begin{gathered} y=(2500)((20)/(13))^(-5) \\ y=290.07 \end{gathered}`

Rounding, because there are no parts of people

`y=291`

Therefore, the population after 5 years of this city will be 291 people.