Science:Math Exam Resources/Courses/MATH102/December 2013/Question C 02 (d)
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Question C 02 (d) 

The population of fish in a particular lake is given by the function F(t) where F is measured in number of fish and t is measured in days. A company that manages fish stocks is hired to restock the lake, adding fish at a constant rate. Only N fishers are allowed to fish in the lake at a time. A simple model for this scenario is given by the equation:
Where I and are constant and two cases for N are considered. Case 2: The agency that administers fishing permits decides that the number of fishers should be proportional to the number of fish currently in the lake, . What is the population size to which F(t) approaches as in this scenario? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

For , the solution reaches its steady state. 
Hint 2 

By plugging in for , the equation becomes

Hint 3 

Because we again search for the steady state of the equation, we set the derivative equal to zero an solve the resulting equation
for . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Applying , the equation becomes For , the solution reaches its steady state . Hence, we again set the lefthand side equal zero and solve for .
We only consider the positive root since the amount of fish can not be negative. 