Science:Math Exam Resources/Courses/MATH102/December 2016/Question A 02

MATH102 December 2016

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Question A 02
A rumour started by one student spreads through UBC according to the model $I'=10I-0.5I^{2},$ in which $I(t)$ denotes the number of students who have heard the rumour by time $t$. Which of the following is the best estimate for the number of students who eventually will hear the rumour? (i) 10 (ii) 200 (iii) 0.5 (iv) 20

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
If the solution to the differential equation will eventually approach some number, it should be a steady state solution.

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. By the hint, we need to find the steady state solutions of the given differential equation, this is when $I'(t)=0$ . $10I-0.5I^{2}=I(10-0.5I)=0\Rightarrow I=0,\ I={\frac {10}{0.5}}=20\Rightarrow \$ $\color {blue}(iv)$