Science:Math Exam Resources/Courses/MATH103/April 2014/Question 05 (b)
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Question 05 (b)
A mutation occurs within a bacterial species that allows it to acquire antibiotic resistance. Through natural selection the abundance of the mutant trait increases in the population. Let be the fraction (i. e. ) of the population, which carries the antibiotic resistance trait. Suppose satisfies the following differential equation:
where is time (years).
At time it is determined that half of the bacterial population carries the antibiotic resistance trait. Determine the fraction of the bacterial population with antibiotic resistance as a function of time .
Work must be shown for full marks. Simplify fully.
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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Use separation of variables to find the general solution to the differential equation.
Use the initial condition to determine the constant in the general solution.
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Use the method of separation of variables. First, re-arrange the differential equation:
Next, integrate both sides. On the right-hand side, we have
where is a constant. On the left-hand side, we have
We can perform the integral, for instance, by substitution. Letting , so that , or equivalently , we get
We have neglected to include the constant because it has already been taken into account above. Thus, we have found that
To determine the constant, use the fact that . Plugging this into the equation above yields
i. e. . Thus,
Solving for gives
Note that we have taken the positive square root as we require the solution to lie in the interval .