Question

The population of Greenview, a small farming town, can be approximated by the model y=9,000(0.87) t, where t is the number of years since year 2017.

a. Does this model represent exponential growth or decay? Explain.
b. By what percent is the population of Greenview changing every year?
c. What was the population of Greenview in 2017?
d. What is the expected population of Greenview in the year 2022? Show work.

Answer 1

The given model represents exponential decay, with an annual percent change of 13%. The population of Greenview in the year 2017 was 9,000. The expected population of Greenview in the year 2022 is approximately 5,899.

Step-by-step explanation:

a. The model y=9,000(0.87)t represents exponential decay because the base of the exponent (0.87) is between 0 and 1. In exponential decay, the quantity decreases over time.

b. To determine the percent change in population every year, we need to find the annual growth rate. The formula for annual growth rate in exponential growth or decay is (1 + r), where r is the rate of change. Therefore, the percent change in population every year is (1 - 0.87) * 100 = 13%.

c. To find the population of Greenview in 2017, we substitute t = 0 into the given model. y = 9,000(0.87)0 = 9,000. Therefore, the population of Greenview in 2017 was 9,000.

d. To find the expected population of Greenview in 2022, we substitute t = 5 into the given model. y = 9,000(0.87)5 ≈ 5,898.58. Therefore, the expected population of Greenview in the year 2022 is approximately 5,899.