Course:MATH102/Question Challenge/2001 December Q5
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Question
In Fish River, the number of salmon (in thousands), , in a given year, is linked to the number of salmon (in thousands), in the following year by the function
where are positive constants.
(a) For what number of salmon is there no change in the number from one year to the next?
(b) Find the number of salmon that would yield the largest number of salmon in the following year.
Hints
Hint 1 |
In the case that there is no change, what can you assume about and ? |
Hint 2 |
The number of salmon in the following year is . You know that is a function of . What should you do to find the largest value of ? |
Solution
Solution to part (a) |
To find the number of salmon that lead to no change the following year, set and solve for . One solution is but to get others, divide through by . Now we have . ...
The carrying capacity of the salmon in Fish River is in thousands of salmon, or salmon. |
Solution to part (b) |
To find the largest value of we will take the first derivative:
To find the value of which grants the maximal value of we set the derivative equal to zero and isolate for :
We know that this value produces a maximum because will be positive for values of less than and negative for values of greater then . The value of which leads to the largest value of is given by . |