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^{(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^{(t)/(3)}`

this is equivalent to:

`2^{(t)/(3)}\text{ = }(5656)/(707)`

this is equivalent to:

`2^{(t)/(3)}\text{ = }8`

this is equivalent to:

`(2^t)^{(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)^{(1)/(3)})^3\text{ = }8^3`

this is equivalent to:

`(2^t)^{(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