Answered

Twenty years ago, a small town in Texas had a population of 10,000. The population has increased by [tex]$8\%$[/tex] each year since then. What equation would be used to solve this population growth problem?

a. [tex]P=20(10,000)^{20}[/tex]
b. [tex]P=10,000(8)^{20}[/tex]
c. [tex]P=10,000(20)^{100}[/tex]
d. [tex]P=10,000(1.08)^{20}[/tex]

Please select the best answer from the choices provided:

A
B
C
D



Answer :

GinnyAnswer
To solve the population growth problem, we need to use the compound growth formula, which can be written as:

[tex]\[ P = P_0 (1 + r)^t \][/tex]

In this formula:
- [tex]\( P \)[/tex] is the population at the end of the period.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the growth rate per period.
- [tex]\( t \)[/tex] is the number of periods.

In our specific problem:
- The initial population ([tex]\( P_0 \)[/tex]) is 10,000.
- The annual growth rate ([tex]\( r \)[/tex]) is 0.08 (or 8\%).
- The number of years ([tex]\( t \)[/tex]) is 20.

Let's plug these values into the formula:

[tex]\[ P = 10,000 (1 + 0.08)^{20} \][/tex]

[tex]\[ P = 10,000 (1.08)^{20} \][/tex]

Therefore, the equation that would be used to calculate the population of the town in 20 years is:

[tex]\[ P = 10,000 (1.08)^{20} \][/tex]

So the best answer from the choices provided is:

D. [tex]\( P = 10,000 (1.08)^{20} \)[/tex]