Science:Math Exam Resources/Courses/MATH102/December 2019/Question 04 (c)

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MATH102 December 2019

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Question 04 (c)

Remember our fishy friends? The quality of hatching of larvae of cold-water fish depends on the temperature of the water. Each hatch is given a quality ranking from 0 to 10, with 10 being a perfect hatch. You want to build a hatchery with Northern Rock Sole (NRS) and Southern Rock Sole (SRS) in separate tanks with temperatures $T_{N}$ and $T_{S}$ respectively. MATH 102 December 2019 Q4c You model the hatch quality of the species as $Q_{N}(T_{N})=10-{\tfrac {1}{20}}(T_{N}-2)^{2}$ for the NRS and $Q_{S}(T_{S})=10-{\tfrac {3}{20}}(T_{S}-8)^{2}$ for SRS. The combined hatch quality is $Q_{N}+Q_{S}$. The equipment you use will make one tank 2$^{\circ }$ cooler than the other. The equipment can handle a temperature range from 0$^{\circ }$ to 16$^{\circ }$ for both $T_{N}$ and $T_{S}$. What should $T_{N}$ and $T_{S}$ be so that they differ by two degrees, both temperatures are in the range $[0,16]$, and the combined quality $Q_{N}+Q_{S}$ is maximized? As always, justify your answer completely, including how you chose which species to be in the cooler tank and which in the warmer.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Maximize the quantity $Q=Q_{N}+Q_{S}=20-{\frac {1}{20}}(T_{N}-2)^{2}-{\frac {3}{20}}(T_{S}-8)^{2}.$ You will need to write $T_{S}$ in terms of $T_{N}$ (or vice-versa) in order to get a single variable to work with.

Hint 2
Hatch quality is higher when the NRS has lower temperature than SRS, so we can assume $T_{S}=T_{N}+2$ and optimize with respect to $T_{N}$.

Hint 3
To justify the optimality of the critical point, use the fact that $Q$ is a quadratic in $T_{N}$ (or $T_{S}$).

Checking a solution serves two purposes: helping you if, after having used all the hints, 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

First, we observe that $Q=Q_{N}+Q_{S}=20-{\frac {1}{20}}(T_{N}-2)^{2}-{\frac {3}{20}}(T_{S}-8)^{2}.$ Hatch quality is higher when the NRS has lower temperature than SRS, so we can assume $T_{S}=T_{N}+2$. Then: ${\begin{aligned}Q(T_{N})&=20-{\frac {1}{20}}(T_{N}-2)^{2}-{\frac {3}{20}}(T_{N}-6)^{2}\\Q'(T_{N})&=-{\frac {1}{10}}(T_{N}-2)-{\frac {3}{10}}(T_{N}-6)\\0&=-{\frac {1}{10}}(T_{N}-2)-{\frac {3}{10}}(T_{N}-6)\\0&=T_{N}-2+3(T_{N}-6)=4T_{N}-2-18\\T_{N}&=5\\T_{S}&=7\end{aligned}}$ Since $Q$ is a parabola pointing down, the sole CP is its absolute maximum. It's in the correct range, so $\color {blue}T_{N}=5$ and $\color {blue}T_{S}=7$ optimize combined hatch quality given the constraints.

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