We have an exponential growth model for the fish population, where P(t) is the population and t is the time in years:
[tex]P(t)=338(3)^{\frac{t}{3}}[/tex]
We have to calculate the time t for which the population is P = 3042.
We start from P(t) = 3042 and rearrange the equation to find the value of t:
[tex]\begin{gathered} P(t)=3042 \\ 338(3)^{\frac{t}{3}}=3042 \\ 3^{\frac{t}{3}}=\frac{3042}{338} \\ 3^{\frac{t}{3}}=9 \\ \frac{t}{3}\ln(3)=\ln(9) \\ \frac{t}{3}=\frac{\ln(9)}{\ln(3)} \\ \frac{t}{3}=2 \\ t=2*3 \\ t=6 \end{gathered}[/tex]
Answer: after t = 6 years, the population will be 3042.
