# Science:Math Exam Resources/Courses/MATH103/April 2014/Question 05 (b)

MATH103 April 2014
Other MATH103 Exams

### 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 ${\displaystyle y}$ be the fraction (i. e. ${\displaystyle 0\leq y\leq 1}$) of the population, which carries the antibiotic resistance trait. Suppose ${\displaystyle y}$ satisfies the following differential equation:

${\displaystyle {\frac {dy}{dt}}=(1-y)^{3},}$

where ${\displaystyle t}$ is time (years).

At time ${\displaystyle t=0}$ 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 ${\displaystyle t}$.

Work must be shown for full marks. Simplify fully.

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