Question

The population of a city is 430​,000 and is increasing at a rate of 4.25​% each year. Approximately when will the population reach 860​,000? ​ (Use a compound growth​ model.)

Answer 1

The population will reach 860,000 in 16.67 years from now.

Explanation:

The compound growth model is given by the following equation:

`P(t) = P(0)(1+r)^(t)`

In which
`P(0)` is the initial population and r is the growth rate(decimal).

In this problem, we have that:

`P(0) = 430000, r = 0.0425`

Approximately when will the population reach 860​,000?

This is t when
`P(t) = 860000`. So

`P(t) = P(0)(1+r)^(t)`

`860000 = 430000(1+0.0425)^(t)`

`(1.0425)^(t) = (860000)/(430000)`

`(1.0425)^(t) = 2`

We have the following logarithm rule

`\log{a^(t)} = t\log{a}`

Applying log to both sides

`\log{(1.0425)^(t)} = \log{2}`

`t\log(1.0425) = 0.3`

`0.018t = 0.3`

`t = (0.3)/(0.018)`

`t = 16.67`

The population will reach 860,000 in 16.67 years from now.