Question

Suppose a population is growing by 8% each year. How many years will it take the population to double?

Answer 1

`\text{9 years}`

Step-by-step explanation

We want to find the number of years that it will take the population to double.

To do this, we have to apply the exponential growth function:

`y=a(1+r)^t`

where y = final value

a = initial value

r = rate of growth

t = time (in years)

For the population to double, it means that the final value must be 2 times the initial value:

`y=2a`

Substitute the given values into the function above:

`\begin{gathered} 2a=a(1+(8)/(100))^t \\ (2a)/(a)=(1+0.08)^t=1.08^t \\ 2=1.08^t \end{gathered}`

To solve further, convert the function from an exponential function to a logarithmic function as follows:

`\log _(1.08)2=t`

Solve for t:

`\begin{gathered} (\log _(10)2)/(\log _(10)1.08)=t \\ \Rightarrow t=9\text{ years} \end{gathered}`

It will take 9 years for the population to double.