Chapter 4.6, Problem 31PE

The population of the United States $P\left(t\right)$ (in millions) since January 1, 1900, can be approximated by $P\left(t\right)=\frac{725}{1+8.295e^{-0.0165t}}$ where t is the number of years since January 1, 1900. (See Example 6)

a. Evaluate $P\left(0\right)$ and interpret its meaning in the context of this problem.

b. Use the function to approximate the U.S. population on January 1, 2020. Round to the nearest million.

c. Use the function to approximate the U.S. population on January 1, 2050.

d. From the model, during which year would the U.S. population reach 500 million?

e. What value will the term $\frac{8.295}{e^{0.0165t}}$ approach as $t\to \infty$?

f. Determine the limiting value of $P\left(t\right)$.