Answer :
To determine the town's population after 4 years given a shrinkage rate of 7% per year, follow these steps:
1. Identify the Initial Population: The starting population of the town is 76,000.
2. Understand the Shrinkage Rate: Each year, the population decreases by 7%, which is equivalent to a multiplier of [tex]\(1 - 0.07 = 0.93\)[/tex].
3. Use the Exponential Decay Formula: The formula to calculate the population after a certain number of years with a constant percentage decrease is:
[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 shrinkage rate (expressed as a decimal).
- [tex]\(t\)[/tex] is the number of years.
4. Substitute the Known Values:
[tex]\[
P = 76,000 \times (0.93)^4
\][/tex]
5. Calculate the Result:
The computation will give us:
[tex]\[
P \approx 56,851.95
\][/tex]
Therefore, after 4 years, the population of the town is approximately 56,852. This equation accurately represents the town's population after 4 years with a 7% annual decrease rate.
1. Identify the Initial Population: The starting population of the town is 76,000.
2. Understand the Shrinkage Rate: Each year, the population decreases by 7%, which is equivalent to a multiplier of [tex]\(1 - 0.07 = 0.93\)[/tex].
3. Use the Exponential Decay Formula: The formula to calculate the population after a certain number of years with a constant percentage decrease is:
[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 shrinkage rate (expressed as a decimal).
- [tex]\(t\)[/tex] is the number of years.
4. Substitute the Known Values:
[tex]\[
P = 76,000 \times (0.93)^4
\][/tex]
5. Calculate the Result:
The computation will give us:
[tex]\[
P \approx 56,851.95
\][/tex]
Therefore, after 4 years, the population of the town is approximately 56,852. This equation accurately represents the town's population after 4 years with a 7% annual decrease rate.
