Question

The population of Waterville increased 12% during 4 years. How many years are required for the population to double its initial value?

Answer 1

24.5 years will be required for the population to double its initial value.

Explanation:

The population of Waterville can be modeled 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.

The population of Waterville increased 12% during 4 years.

This means that
`P(4) = 1.12P_(0)`

With this, we can find r

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

`1.12P_(0) = P_(0)(1 + r)^(4)`

`(1+r)^(4) = 1.12`

Applying the fourth root to both sides

`1 + r = 1.0287`

So

`P(t) = P_(0)(1.0287)^(t)`

How many years are required for the population to double its initial value?

This is t when
`P(t) = 2P_(0)`

So

`P(t) = P_(0)(1.0287)^(t)`

`2P_(0) = P_(0)(1.0287)^(t)`

`(1.0287)^(t) = 2`

How we find t?

Logarithims

We have that

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

So we apply log to both sides

`\log{(1.0287)^(t)} = \log{2}`

`t\log{1.0287} = \log{2}`

`t = \frac{\log{2}}{\log{1.0287}}`

`t = 24.5`

24.5 years will be required for the population to double its initial value.