Question

A certain species of alligators is to be introduced into a swamp, and wildlife experts estimate the population will grow to P(t) = (790)3ᵗ/4 , where t represents the number of years from the time of introduction. What is the tripling-time for this population of alligators?

Answer 1

The tripling-time for the population of alligators is determined by solving for the value of t when the initial population, given by P(t) = (790)3^(t/4), has tripled. The calculation reveals that the tripling-time for this alligator population is 4 years.

Step-by-step explanation:

The question asks for the tripling-time for a population of alligators given their growth model P(t) = (790)3t/4.

To solve this mathematical problem completely, we need to determine the value of t at which the initial population has tripled in size. Tripling the initial population of 790 alligators gives us a target population of P(t) = 790 × 3.

Set the population function equal to the target population and solve for t:

1. P(t) = (790)3t/4 = 2370
2. Divide both sides by 790 to isolate the exponential term:
   3t/4 = 3
3. Apply logarithms to solve for t:
   t/4 × log(3) = log(3), and therefore t = 4.

Thus, the tripling-time for this population of alligators is 4 years.