Question

For 10 years, the population of a city increases in a pattern that is approximately exponential. Using exponential modeling you find a function p(t) to describe the population where p is the population in thousands) of the city in year t. Your function uses an initial value of 15 and a 6.4% annual growth rate. What is a reasonable prediction for the future population of the city? A The population will be under 7000 in Year 12. The population will be over 40,000 in Year 15. The population will be under 50,000 in Year 18. D The population will be over 300,000 by Year 20.

Answer 1

We can use the following equation to describe the exponential function:

`P=P_0\cdot(1+i)^t`

Where P is the final value after t years, P0 is the initial value and i is the growth rate per year.

Using P0 = 15 and i = 6.4% = 0.064, we have:

`P=15(1.064)^t`

Now let's check each option:

A) For t = 12 we have:

`\begin{gathered} P=15(1.064)^(12) \\ P=31.578 \end{gathered}`

The population is approximately 31,578, so this option is incorrect.

B) For t = 15 we have:

`\begin{gathered} P=15(1.064)^(15) \\ P=38.038 \end{gathered}`

This option is also incorrect, the population is under 40,000.

C) For t = 18:

`\begin{gathered} P=15(1.064)^(18) \\ P=45.818 \end{gathered}`

The population is under 50,000, so this option is correct.

D) For t = 20:

`\begin{gathered} P=15(1.064)^(20) \\ P=51.871 \end{gathered}`

The population is not over 300,000, so this option is incorrect.

The correct option is C.