Question

A rabbit farm had 200 rabbits in 2015. the number of rabbits increases by 30% every year. how long will it take for the rabbit population to reach 1,000?

Answer 1

`Approximately\:6\:years`

1) Since the initial population of rabbits consists of 200 animals and there is a constant growth of 30% so we can start out writing the exponential model for this problem, like this.

`P=200(1+0.3)^t`

2) Let's plug P=1000 for the final population and solve for t:

`1000=200(1+0.3)^t`

So, let's count 2015 as our first year since we don't know when there will be 1,000 rabbits.

`\begin{gathered} 1000=200(1.3)^t \\ (1000)/(200)=(200(1.3)^t)/(200) \\ 5=(1.3)^t \\ \ln(5)=\ln(1.3)^t \\ \ln \left(5\right)=t\ln \left(1.3\right) \\ t=(\ln \left(5\right))/(\ln \left(1.3\right)) \\ t=6.13\approx6 \end{gathered}`

Note the property of the exponents of logarithms