Science:Math Exam Resources/Courses/MATH110/April 2014/Question 08 (a)

MATH110 April 2014

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Question 08 (a)

Imagine bushbuck antelopes interacting in a montane forest. Let $P$ denote the probability that a given antelope is in contact with another antelope. $P$ is modelled by the equation $P=1-e^{-\pi \rho D^{2}},$ where $D$ is a constant denoting the corresponding distances of the species, and $\rho$ is a function denoting the density of antelope in the area. Suppose $\rho '(t)$ is determined by field observations. Come up with an expression for the rate of change of the probability that an individual is in contact with another individual at time $t$. (Your answer will involve the term $\rho '(t)$.)

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
Start by identifying which symbols in the equations are constants and which ones change over time.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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. We make the assumption that $\rho$ is the only parameter that changes with time. So we get: ${\begin{aligned}P(t)&=1-e^{-\pi \rho (t)D^{2}}\\P'(t)&=\pi D^{2}\rho '(t)e^{-\pi \rho (t)D^{2}}\end{aligned}}$

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