High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ A town has a population of 126,000 and shrinks at a rate of [tex]4.3\%[/tex] every year. Which equation represents the town's population after 2 years?

A. [tex]P=126,000(1.043)^2[/tex]
B. [tex]P=126,000(1-0.043)(1-0.043)[/tex]
C. [tex]P=126,000(0.957)(0.957)[/tex]
D. [tex]P=126,000(0.043)^2[/tex]

Answer :

Let's start by understanding how population shrinkage works year over year. The town's population decreases by 4.3% each year.

1. Initial population is [tex]\( P_0 = 126,000 \)[/tex].
2. The shrinkage rate is 4.3%, which can be written as [tex]\(\text{shrink rate} = 0.043\)[/tex].

To find the population after one year, we calculate:
[tex]\[
P_1 = P_0 \times (1 - \text{shrink rate})
\][/tex]
[tex]\[
P_1 = 126,000 \times (1 - 0.043)
\][/tex]
[tex]\[
P_1 = 126,000 \times 0.957
\][/tex]
[tex]\[
P_1 = 120,582.0
\][/tex]

To find the population after the second year, we apply the shrinkage rate again to [tex]\( P_1 \)[/tex]:
[tex]\[
P_2 = P_1 \times (1 - \text{shrink rate})
\][/tex]
[tex]\[
P_2 = 120,582 \times 0.957
\][/tex]
[tex]\[
P_2 = 115,396.974
\][/tex]

Alternatively, we can also find the population after 2 years using the formula:
[tex]\[
P = P_0 \times (1 - \text{shrink rate})^2
\][/tex]
Plugging in the values:
[tex]\[
P = 126,000 \times (1 - 0.043)^2
\][/tex]
[tex]\[
P = 126,000 \times 0.957^2
\][/tex]

When we calculate [tex]\( 0.957^2 \)[/tex]:
[tex]\[
0.957^2 = 0.915849
\][/tex]

Thus, the population after 2 years is:
[tex]\[
P = 126,000 \times 0.915849
\][/tex]
[tex]\[
P = 115,396.97399999999
\][/tex]

Given the options, the correct equation representing the town's population after 2 years is

[tex]\[
P = 126,000 \times 0.957 \times 0.957
\][/tex]

This equation matches:
[tex]\[
P = 126,000 \times (0.957)^2
\][/tex]
The equivalent choice is:

[tex]\[
P = 126,000 (0.957)(0.957)
\][/tex]