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

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 ${\displaystyle T_{N}}$ and ${\displaystyle T_{S}}$ respectively.

• You model the hatch quality of the species as ${\displaystyle Q_{N}(T_{N})=10-{\tfrac {1}{20}}(T_{N}-2)^{2}}$ for the NRS and ${\displaystyle Q_{S}(T_{S})=10-{\tfrac {3}{20}}(T_{S}-8)^{2}}$ for SRS.
• The combined hatch quality is ${\displaystyle Q_{N}+Q_{S}}$.
• The equipment you use will make one tank 2${\displaystyle ^{\circ }}$ cooler than the other.
• The equipment can handle a temperature range from 0${\displaystyle ^{\circ }}$ to 16${\displaystyle ^{\circ }}$ for both ${\displaystyle T_{N}}$ and ${\displaystyle T_{S}}$.

What should ${\displaystyle T_{N}}$ and ${\displaystyle T_{S}}$ be so that they differ by two degrees, both temperatures are in the range ${\displaystyle [0,16]}$, and the combined quality ${\displaystyle Q_{N}+Q_{S}}$ 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.

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