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

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.

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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.

• 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.