Science:Math Exam Resources/Courses/MATH103/April 2016/Question 06 (a) (v)
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Question 06 (a) (v) 

Consider a population that is governed by the logistic differential equation with initial condition . is the population size and is time measured in years. (v) How long will it take for the population to reach 50% of the carrying capacity? 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Use the results in part (ii) and part (iv). 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. From parts (ii) and (iv), we have that if and then Let be the time at which the population reaches 50% of the carrying capacity. Then, at time we know that and therefore Thus, we solve the following equation to find : So 