Question

The population, P, of a certain town can be modeled by the equation P=2005e^kt, where t=0 represents the year 1990.

a. If the population of the town in 1975 was 2,183, find the growth rate, k, of the town.
b. What does the model estimate this town's population to be in the year 2020?

Answer 1

To find the growth rate k for a town's population modeled by an exponential equation, we use historical population data and solve for k. Then, we can use this growth rate to estimate the population in a future year by substituting the year difference into the model.

Step-by-step explanation:

The population, P, of a certain town can be modeled by the equation P=2005ekt, where t=0 represents the year 1990.

a. To find the growth rate, k, we use the information that the population of the town in 1975 was 2,183. Since 1975 is 15 years before 1990, we set t to -15 and solve for k:

2183 = 2005ek(-15)

Dividing both sides by 2005 and taking the natural logarithm, we get:

ln(2183/2005) = -15k

k = ln(2183/2005) / -15

b. To estimate the town's population in the year 2020, since 2020 is 30 years after 1990, we set t to 30 and use the value of k obtained from part a:

P = 2005ek(30)

This equation will give us the modeled population in 2020.