Over a period of 20 years, the population P (in thousands) of a town can be modeled

Question

Over a period of 20 years, the population P (in thousands) of a town can be modeled by P = 0.0081t^3 − 0.245t^2 + 2.13t + 95.8 where t is the time (in years).

a. Use a graphing calculator to graph the function for the interval 1 ≤ t ≤ 20. Describe the behavior of the graph on this interval.

From around Year ? and from around Year ? , the town's population increased. From around Year ? to Year ?, the population decreased.

b. What was the average rate of change in the town's population from Year 1 to Year 20? Round your answer to the nearest tenth.

The average rate of change was about ? thousand people per year.

c. Use the model to predict the town's population in Year 25. Round your answer to the nearest tenth. Population in Year 25: ? thousand people

d. Why should the model not be used for many years after Year 20?

A. The model population approaches negative infinity as the year approaches infinity.

B. The model population approaches infinity as the year approaches infinity.

C. The model population stays the same as the year approaches infinity.

Answer 1

Don't have graphing calculator, so whipped up the numbers with Excel. A cubic function is clear with local maximum around 6.4 years and local minimum around t = 13.6 years.

a) Numbers increased from t = 0 to t = 6, and again from t =14 to t = 20. Numbers decreased from t = 7 to t = 13.

b) Average rate of change = [P(20) - P(0)] / Δt = (105.20 - 95.8)/20 = 0.470 thousand people per year, or 470 people per year.

c) At t = 25, P = 122.5 thousand.

d) B (it's a cubic after all with positive leading term).