Question

The population of a city is modeled by the exponential function ƒ (x​) = 689,254 · 2.40.014x​, where x is the number of years after 2000. In what year will the city's population exceed one million?

Answer 1

2036

Explanation:

Given the population of a city is modeled by the exponential function

ƒ (x​) = 689,254 · 2.4e^0.014x​, where x is the number of years after 2000

In order to determine the year that the city's population will exceed one million, we will substitute f(x) = 1,000,000 and find x first as shown:

1,000,000 = 689,254 2.4e^0.014x​,

1000000/689,254 = 2.4e^0.014x​,

1.4508 = 2.4e^0.014x​,

1.4508 /2.4 = e^0.014x​,

0.6045 = e^0.014x​,

Apply ln to both sides

ln 0.6045 = lne^0.014x​,

-0.5033 = 0.014x

x = -35.95

Since x cannot be negative, lets say x ≈ 36years

This means that the population will exceed one million after about 36 years. The year it will exceed this amount will be 2000+36 = 2036

Note that the exponential equation used was not correctly written but the same calculation can be employed for any exponential functions you might have.