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^{(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^{(0)/(2)}=(674)3^0=674*1=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^{(t)/(2)} \\ \Rightarrow3^{(t)/(2)}=(2022)/(674)\Rightarrow3^{(t)/(2)}=3 \\ \Rightarrow3^{(t)/(2)}=3^1\Rightarrow(t)/(2)=1 \\ \Rightarrow t=2*1=2 \end{gathered}`

Hence, the tripling time is 2 years.