Question

In the year 2000, the United States had a population of about 281.4 million people; by 2010, the population had risen to about 308.7 million.

Part A

Find the 10-year continuous growth rate using P=
`P_(0)`
`e^(rt)`.


Part B

Write an equation to model the population growth of the United States, and use it to estimate the population in 2020.

Answer 1

Part 1. 0.9259 % per year

Part 2. P = 281.4e^(0.009 259t); 338.6 million

Explanation:

Data:

P₀ = 281.4 million

P = 308.7 million

Part 1. Growth rate

t = 2010 - 2000 = 10 yr

P = P₀e^(rt)

308.7 = 281.4e^(10r)

e^(10r) = 1.0970

10r = ln1.0970

r = (ln1.0970)/10 = (0.092 59)/10 = 0.009 259

r = 0.9259 % per year

The 10-year continuous growth rate is 0.9259 % per year.

Part 2. Population model

The population model is

P = 281.4e^(0.009 259t)

where P is in millions and t is the number of years since 2000.

By 2020,

P = 281.4e^(0.009 259 × 20) = 281.4e^0.1852 = 281.4 × 1.203

P = 338.6 million

The estimated population in 2020 is 338.6 million.