OSH/3

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It is the final exam period and you have only math and chemistry exams left. They are on the same day and you have 20 hours of study time remaining. You estimate that your mark on the chemistry exam as a function of hours spent studying is

$C(x)=100\dfrac{x}{4+x}.$

You can't be quite as certain about your mark in your math course but you know it has a similar form:

$M(x)=100\dfrac{x}{k+x}.$

where $k>0$.

Submit parts 1-4 as Q1 on Crowdmark:

  1. For this part only, suppose that $k>4$. (a) Interpret what this means about the comparative difficulty of studying math and chemistry. (b) Without doing any calculations, do you think you should spend more time studying for math or chemistry? Give an honest answer based on your thoughts before doing the rest of this OSH - part (b) will not be marked for correctness, it is intended to encourage you to build an expectation that you can compare later to your formal answer - a valuable step in problem solving.
  2. Write down a function for your average mark $A(x)$ on the two exams where $x$ is the number of hours spent studying math. Assume the remaining time is spent entirely on chemistry. Write your answer in terms of the functions $C$ and $M$.
  3. What is (a) the mathematical domain of the model and (b) the model domain? That is, what is the set of $x$ values that can be plugged into $A(x)$ in both (a) a mathematical sense and (b) a realistic sense given the meaning of $x$ in the problem?
  4. Verify by calculating them from scratch that there are two critical points of $A(x)$ given by $x_1=\dfrac{2\sqrt{k}(12-\sqrt{k})}{\sqrt{k}+2}$ and $x_2=\dfrac{2\sqrt{k}(12+\sqrt{k})}{\sqrt{k}-2}$. Hint: it is easiest to proceed as far as possible with $A(x)$ written in terms of $C$ and $M$ before substituting in the algebraic expressions for these functions (or more accurately, the algebraic expressions for their derivatives).

Submit parts 5-7 as Q2 on Crowdmark:

  1. For what values of $k$ is $x_1$ in the domain of the model? For what values of $k$ is $x_2$ in the domain of the model?
  2. Calculate $M''(x)$ and $C''(x)$. Use these to calculate $A''(x)$. For any value of $x$ between 0 and 20, what is the sign of $A''(x)$? If, for some value $0<x_*<20$, you found that $A'(x_*)=0$, what can you conclude about $x_*$?
  3. For what values of $k$ is it best (in the sense of largest value of $A(x)$) to spend more time studying math than chemistry? You must show your calculations and justify the claim that you are considering the best rather than the worst option.

Submit parts 8-9 as Q3 on Crowdmark:

  1. Sketch a graph of the optimal amount of time to spend studying math as a function of the parameter $k$. Your $k$ axis should extend from 0 to 200. You do not have to justify the shape of your graph - in principle, you could use calculus techniques to graph it but those calculations are very messy so you can do it by any means you choose.
  2. Explain in words why this model predicts that it is best (again, in the sense of largest value of $A(x)$) to spend the majority of your study time on chemistry when math is much easier than chemistry AND when math is much harder than chemistry. Comment on your initial expectation from part 1.