Question

A certain species of alligators is to be introduced into a swamp, and wildlife experts estimate the population will grow toP(t) = (674)3^(t/2), where t represents the number of years from the time of introduction.What is the tripling-time for this population of alligators?

Answer 1

$P(t)=(674)3^{\frac{t}{2}}$

To find the tripling time, you have to calculate the time when the population will triple.

First, calculate the initial population of the animal by substituting t=0 into the function:

$P(0)=(674)3^{\frac{0}{2}}=(674)3^0=674\times1=674$

It follows that the population triples when P(t)=674×3=2022.

Substitute this value into the function and solve for t:

$\begin{gathered} 2022=(674)3^{\frac{t}{2}} \\ \Rightarrow3^{\frac{t}{2}}=\frac{2022}{674}\Rightarrow3^{\frac{t}{2}}=3 \\ \Rightarrow3^{\frac{t}{2}}=3^1\Rightarrow\frac{t}{2}=1 \\ \Rightarrow t=2\times1=2 \end{gathered}$

Hence, the tripling time is 2 years.