A town has a population of 8400 in 1990. Fifteen years later, it's population grew to 125,000. Determine the average annual growth rate of this town's population. Round to the nearest tenth of a percent.
To determine the average annual growth rate of the town's population, we need to find the rate at which the population grew over a period of 15 years. We can use the formula for compound interest to calculate this.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = final amount (population after 15 years)
P = initial amount (population in 1990)
r = annual interest rate (growth rate)
n = number of times the interest is compounded per year (for simplicity, we'll assume it's compounded annually)
t = number of years
In this case, we know the initial population (P = 8400), the final population after 15 years (A = 125000), and the number of years (t = 15). We need to solve for the annual interest rate (r).
125000 = 8400(1 + r/1)^(1*15)
Simplifying the equation:
(1 + r)^(15) = 125000 / 8400
Taking the 15th root of both sides:
1 + r ≈ (125000 / 8400)^(1/15)
1 + r ≈ 1.07151962
Subtracting 1 from both sides:
r ≈ 0.07151962
To express this as a percentage, we multiply by 100:
r ≈ 7.15%
Therefore, the average annual growth rate of this town's population is approximately 7.15%.
125,000 - 8,400 = 116,600
116,600 / 15 = 7,773.33333
7,773.33 / 8,400 = 0.925396429 = 92.5% annual growth