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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.
[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.
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