007B Sample Midterm 1, Problem 2 Detailed Solution

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A population grows at a rate

where    is the population after    months.

(a)   Find a formula for the population size after    months, given that the population is    at  

(b)   Use your answer to part (a) to find the size of the population after one month.

Background Information:  



Step 1:  
To find    we need to integrate  .
We get
Now, we use  -substitution.
Let    Then,  
So, we have
Step 2:  
Now, we need to find the constant  
From the information provided in the problem, we know  
So, we have
Solving for    we get  
Hence, we have

We need to calculate  
Using our answer from part (a), we have


Final Answer:  

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