Answer :
Certainly! To solve this problem, we need to calculate the future population of the town based on exponential growth, using the formula for compound interest.
The formula for compound growth is:
[tex]\[ P = P_0 \times (1 + r)^t \][/tex]
where:
- [tex]\( P \)[/tex] is the future population.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the annual growth rate.
- [tex]\( t \)[/tex] is the number of years.
Given:
- Initial population ([tex]\( P_0 \)[/tex]) = 14,000
- Annual growth rate ([tex]\( r \)[/tex]) = 0.05 (which is 5%)
- Number of years ([tex]\( t \)[/tex]) = 14
Let's calculate step-by-step:
1. Initial Population: [tex]\( P_0 = 14,000 \)[/tex]
2. Growth Rate: [tex]\( r = 0.05 \)[/tex]
3. Number of Years: [tex]\( t = 14 \)[/tex]
Plug these values into the formula:
[tex]\[ P = 14,000 \times (1 + 0.05)^{14} \][/tex]
First, calculate [tex]\( 1 + 0.05 \)[/tex]:
[tex]\[ 1 + 0.05 = 1.05 \][/tex]
Next, raise 1.05 to the power of 14:
[tex]\[ 1.05^{14} \approx 1.979 \][/tex]
Now, multiply this result by the initial population:
[tex]\[ P \approx 14,000 \times 1.979 \][/tex]
[tex]\[ P \approx 27,706 \][/tex]
Hence, the population after 14 years, rounded to the nearest whole number, will be approximately 27,706.
The formula for compound growth is:
[tex]\[ P = P_0 \times (1 + r)^t \][/tex]
where:
- [tex]\( P \)[/tex] is the future population.
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the annual growth rate.
- [tex]\( t \)[/tex] is the number of years.
Given:
- Initial population ([tex]\( P_0 \)[/tex]) = 14,000
- Annual growth rate ([tex]\( r \)[/tex]) = 0.05 (which is 5%)
- Number of years ([tex]\( t \)[/tex]) = 14
Let's calculate step-by-step:
1. Initial Population: [tex]\( P_0 = 14,000 \)[/tex]
2. Growth Rate: [tex]\( r = 0.05 \)[/tex]
3. Number of Years: [tex]\( t = 14 \)[/tex]
Plug these values into the formula:
[tex]\[ P = 14,000 \times (1 + 0.05)^{14} \][/tex]
First, calculate [tex]\( 1 + 0.05 \)[/tex]:
[tex]\[ 1 + 0.05 = 1.05 \][/tex]
Next, raise 1.05 to the power of 14:
[tex]\[ 1.05^{14} \approx 1.979 \][/tex]
Now, multiply this result by the initial population:
[tex]\[ P \approx 14,000 \times 1.979 \][/tex]
[tex]\[ P \approx 27,706 \][/tex]
Hence, the population after 14 years, rounded to the nearest whole number, will be approximately 27,706.
