Population Math Problem

Question

According to the U. S. Bureau of the Census, in 2000 there were 35.3e million residents of Hispanic origin living in the United States. By 2010, the number had increased to 50.5 million. The exponential growth function A = 35.3e^kt describes the U.S. Hispanic population, A, in millions, t years after 2000.

a. Find k, correct to three decimals places.

b. Use the resulting model to project the Hispanic resident in population in 2015.

c. In which year will the Hispanic resident population reach 70 million?

Answer 1

The key to all of the parts is using natural logarithms to isolate and solve for the kt exponent terms.

Part a: A = 50.5

t = 10 years

k = ?

Math steps:

A = 35.3e^kt

A = e^kt

35.3

ln(A/35.3) = ln(e^kt)

ln(A/35.3) = kt

k = ln(A/35.3)

t

k = ln(50.5/35.3)

10

k = 0.036

Part b: A = ?

t = 15 years

k = 0.036

Math steps:

A = 35.3e^kt

= 35.3e^(0.036*15)

A = 60.6

Part c: A = 70

t = ?

k = 0.036

Math steps:

A = 35.3e^kt

A = e^kt

35.3

ln(A/35.3) = ln(e^kt)

ln(A/35.3) = kt

t = ln(A/35.3)

k

t = ln(70/35.3)

0.036

t = 19 years

2000 + 19 = 2019

Population will be 70 million by 2019