The population of a town was 5655 in 2010. The population grows at a rate of 1.4% annually.
Use the exponential growth model to write an equation that estimates the population t years after 2010.
Estimate the population of the town in 2022. Show your work.
To estimate the population t years after 2010 using the exponential growth model, we can use the formula:
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 as a decimal
t is the number of years after the initial time
Given that the initial population in 2010 was 5655 and the growth rate is 1.4% annually (which is equivalent to 0.014 as a decimal), we can write the equation as:
P(t) = 5655 * (1 + 0.014)^t
To estimate the population of the town in 2022, we need to calculate the number of years from 2010 to 2022. This is 12 years since 2022 is 12 years after 2010.
Substituting t = 12 into the equation, we can calculate:
P(12) = 5655 * (1 + 0.014)^12
P(12) = 5655 * (1.014)^12
Calculating this expression:
P(12) ≈ 5655 * (1.014)^12 ≈ 5655 * 1.190927 ≈ 6747.58
Therefore, the estimated population of the town in 2022 is approximately 6748.
To estimate the population t years after 2010 using the exponential growth model, we can use the formula:
P(t) = P(0) * (1 + r)^t
Where:
P(t) = the population t years after 2010
P(0) = the initial population in 2010
r = the growth rate
t = the number of years after 2010
In this case, the initial population in 2010 is 5655, and the growth rate is 1.4% or 0.014 (in decimal form).
Therefore, the equation that estimates the population t years after 2010 is:
P(t) = 5655 * (1 + 0.014)^t
To estimate the population of the town in 2022, we need to find t, the number of years after 2010.
2022 - 2010 = 12
So, t = 12.
Now, substitute t = 12 into the equation:
P(12) = 5655 * (1 + 0.014)^12
Evaluate the right side using a calculator:
P(12) ≈ 5655 * (1.014)^12
P(12) ≈ 5655 * 1.191016
P(12) ≈ 6731.88
Therefore, the estimated population of the town in 2022 is approximately 6732.
Please note that this is an estimate and the actual population may vary.