Question:
Solution:
The population growth is given by the following equation:
[tex]P(t)=(707)2^{\frac{t}{3}}[/tex]
where P represents the number of individuals and t represents the number of years from the time of introduction. Now, if we have a population of 5656 fish, then the above equation becomes:
[tex]5656=(707)2^{\frac{t}{3}}[/tex]
this is equivalent to:
[tex]2^{\frac{t}{3}}\text{ = }\frac{5656}{707}[/tex]
this is equivalent to:
[tex]2^{\frac{t}{3}}\text{ = }8[/tex]
this is equivalent to:
[tex](2^t)^{\frac{1}{3}}\text{ = }8[/tex]
now, the inverse function of the root function is the exponential function. So that, we can apply the exponential function to the previous equation:
[tex]((2^t)^{\frac{1}{3}})^3\text{ = }8^3[/tex]
this is equivalent to:
[tex](2^t)^{\frac{3}{3}}^{}\text{ = }512[/tex]
this is equivalent to:
[tex]2^t\text{ = }512[/tex]
now, we can apply the properties of the logarithms to the previous equation:
[tex]\log _2(2^t)\text{ = }log_2(512)[/tex]
this is equivalent to:
[tex]t=log_2(512)\text{ = 9}[/tex]
we can conclude that the correct answer is:
9 years
