High School

A town has a population of 76,000 and shrinks at a rate of [tex]7\%[/tex] every year. Which equation represents the town's population after 4 years?

A. [tex]P = 76,000(1-0.07)^4[/tex]
B. [tex]P = 76,000(1-0.07)(1-0.07)(1-0.07)(1-0.07)[/tex]
C. [tex]P = 76,000(0.07)^4[/tex]
D. [tex]P = 76,000(1.07)^4[/tex]

Answer :

To solve the problem of finding the town's population after 4 years when it shrinks at a rate of 7% per year, you can use the formula for exponential decay:

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

Where:
- [tex]\( P \)[/tex] is the population after [tex]\( t \)[/tex] years.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the rate of decay (as a decimal).
- [tex]\( t \)[/tex] is the time in years.

In this problem:
- [tex]\( P_0 = 76,000 \)[/tex]
- [tex]\( r = 0.07 \)[/tex] (since 7% as a decimal is 0.07)
- [tex]\( t = 4 \)[/tex] (since we want to find the population after 4 years)

Plug these values into the formula:

[tex]\[ P = 76,000 \times (1 - 0.07)^4 \][/tex]

[tex]\[ P = 76,000 \times (0.93)^4 \][/tex]

When you carry out the calculation of [tex]\( (0.93)^4 \)[/tex], you'll find:

[tex]\[ (0.93)^4 = 0.7529536 \][/tex]

Now multiply this result by the initial population:

[tex]\[ P = 76,000 \times 0.7529536 \][/tex]

[tex]\[ P \approx 56,851.95 \][/tex]

So, the town's population after 4 years is approximately 56,851.