Question

how long will it take for the population to reach 5656 fish, according to this model?

Answer 1

Question:

Solution:

The population growth is given by the following equation:

$P(t)=(707)2^{\frac{t}{3}}$

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:

$5656=(707)2^{\frac{t}{3}}$

this is equivalent to:

$2^{\frac{t}{3}}\text{ = }\frac{5656}{707}$

this is equivalent to:

$2^{\frac{t}{3}}\text{ = }8$

this is equivalent to:

$(2^t)^{\frac{1}{3}}\text{ = }8$

now, the inverse function of the root function is the exponential function. So that, we can apply the exponential function to the previous equation:

$((2^t)^{\frac{1}{3}})^3\text{ = }8^3$

this is equivalent to:

$(2^t)^{\frac{3}{3}}^{}\text{ = }512$

this is equivalent to:

$2^t\text{ = }512$

now, we can apply the properties of the logarithms to the previous equation:

$\log _2(2^t)\text{ = }log_2(512)$

this is equivalent to:

$t=log_2(512)\text{ = 9}$

we can conclude that the correct answer is:

9 years