009B Sample Final 3, Problem 3

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The population density of trout in a stream is

where    is measured in trout per mile and    is measured in miles.    runs from 0 to 12.

(a) Graph    and find the minimum and maximum.

(b) Find the total number of trout in the stream.

What is the relationship between population density    and the total populations?
       The total population is equal to  
       for appropriate choices of  



Step 1:  
To graph    we need to find out when    is negative.
To do this, we set
So, we have
Hence, we get    and   
But,    is outside of the domain of  
Using test points, we can see that    is positive in the interval  
and negative in the interval  
Hence, we have
The graph of    is displayed below.
Step 2:  
We need to find the absolute maximum and minimum of  
We begin by finding the critical points of
Taking the derivative, we get
Solving    we get a critical point at
Now, we calculate  
We have
Therefore, the minimum of    is    and the maximum of    is  


Step 1:  
To calculate the total number of trout, we need to find
Using the information from Step 1 of (a), we have
Step 2:  
We integrate to get
Thus, there are approximately    trout.

Final Answer:  
    (a)     The minimum of    is    and the maximum of    is   (See above for graph.)
    (b)     There are approximately    trout.

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