Science:Math Exam Resources/Courses/MATH102/December 2015/Question 15
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Question 15 

Someone released a group of 10 rabbits on campus and they started reproducing at a rate proportional to the population size with constant of proportionality of 3 per year. After 5 years, the rabbits continued to reproduce in the same manner but a coyote moved in and began eating rabbits at a rate proportional to the population size with a constant of proportionality of 9 per year. The population of rabbits is given by the function
Give values for , , , and . Your answers can be in terms of exponentials and/or logarithms. 
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? 
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Hint 

Write the given information in terms of . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Someone released a group of 10 rabbits on campus and they started reproducing at a rate proportional to the population size with constant of proportionality of 3 per year. After 5 years, the rabbits continued to reproduce in the same manner but a coyote moved in and began eating rabbits at a rate proportional to the population size with a constant of proportionality of 9 per year. The population of rabbits is given by the function
According to the question, at the initial time (), we have 10 rabbits, so that Until 5 years, the population increases at a rate proportional to the population size with constant of proportionality of 3 per year. This implies that for . By using these two equations, we can find and in the given formula on the interval ; .
On the other hand, since the rabbits are eaten by the coyote at a rate proportional to the population size with a constant of proportionality of 9 per year, the rate of change of the population of rabbits becomes
Now, as and are found, we can get and ; from the given formula on ,
To summarize, we obtain . 