007B Sample Midterm 3, Problem 3 Detailed Solution

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For a fish that starts life with a length of 1cm and has a maximum length of 30cm, the von Bertalanffy growth model predicts that the growth rate is    cm/year where    is the age of the fish. What is the average length of the fish over its first five years?

Background Information:  
The average value of a function    on an interval    is given by


Step 1:  
Let    be the length of the fish at age  
Given the information in the problem, we know
First, we find   
We have


Step 2:  
Next, we need to solve for  
We have
So, we get  


Step 3:  
Finally, we need to find the average value of    over the interval  
We have

Final Answer:  

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