Question

A city has 180,000 residents. The population is increasing at the rate of 9% per year. Make a prediction for how long it will take for the population to reach 300,000. Round your answer to the nearest tenth of a year. The population will reach 300,000 in approximately how many years.

Answer 1

1.09

Explanation:

Answer 2

In one year, the population will increase by 9% of its original size, that is,
`180,000(0.09) + 180,000 = 180,000(0.09 + 1) = 180,000(1.09)`, which is the population multiplied by 1.09.

The next year, the population will be again multiplied by 1.09:
`180,000(1.09)(1.09)`.

We can write the trend as an equation:
P = 180000(1.09)^n, where n is the number of years and P is the population after n years.

To find the number of years for for the population to reach 300,000, we can substitute 300,000 into our equation and solve for n.

`P = 180000^n \\ 300000 = 180000(1.09)^n \\ (5)/(3) = 1.09^n \\ \log_(1.09) (5)/(3) = n \\ \\ n = 5.9`