A town has a population of 8400 in 1990. Fifteen years later, its population grew to 12500. Determine the average annual growth rate of this town's population. How do i solve this using applications of exponential functions?
12500=8400(R)^15
ln12500=ln8400+15lnR
solve for R
it doesnt work,
the answer is approximately 2.7%
:(
To determine the average annual growth rate of the town's population using applications of exponential functions, you can follow these steps:
Step 1: Understand the formula for exponential growth. The formula for exponential growth is given by:
P(t) = P₀ * (1 + r)^t
where:
P(t) = population after time t
P₀ = initial population
r = growth rate
t = time in years
Step 2: Use the given information to form two equations. Let's say P₀ = 8400 (population in 1990) and P(t) = 12500 (population after 15 years).
Equation 1: 12500 = 8400 * (1 + r)^15
Equation 2: P(t) = P₀ * (1 + r)^t
Step 3: Solve Equation 1 for r. Since it is given that P₀ = 8400 and P(t) = 12500 after 15 years, substitute these values into Equation 1 and solve for r.
12500 = 8400 * (1 + r)^15
(1 + r)^15 = 12500 / 8400
(1 + r)^15 = 1.4881
Step 4: Take the 15th root of both sides of the equation to isolate (1 + r).
1 + r = (1.4881)^(1/15)
1 + r ≈ 1.047
Step 5: Subtract 1 from both sides to isolate r.
r ≈ 1.047 - 1
r ≈ 0.047
Step 6: Convert the growth rate to a percentage.
Growth rate ≈ 0.047 * 100%
Growth rate ≈ 4.7%
Therefore, the average annual growth rate of the town's population is approximately 4.7%.
To determine the average annual growth rate of the town's population using exponential functions, you need to use the formula for exponential growth:
P(t) = P(0) * (1 + r)^t
Where:
- P(t) is the population at time t
- P(0) is the initial population
- r is the growth rate
- t is the time period in years
In this case, let's consider 1990 as the starting point (t = 0) with a population of 8400. We also know that after 15 years (t = 15), the population grew to 12500.
Substituting the given values into the formula:
12500 = 8400 * (1 + r)^15
To solve for the growth rate, we can rewrite the equation as:
(1 + r)^15 = 12500 / 8400
Now, we need to find the 15th root of both sides to isolate (1 + r):
(1 + r) = (12500 / 8400)^(1/15)
Finally, subtract 1 from both sides to get the growth rate (r):
r = (12500 / 8400)^(1/15) - 1
Plug this equation into a calculator or a spreadsheet to compute r. The resulting value will be the average annual growth rate of the town's population.