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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 and respectively.
What should and be so that they differ by two degrees, both temperatures are in the range , and the combined quality 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.
Maximize the quantity
You will need to write in terms of (or vice-versa) in order to get a single variable to work with.
Hatch quality is higher when the NRS has lower temperature than SRS, so we can assume and optimize with respect to .
To justify the optimality of the critical point, use the fact that is a quadratic in (or ).
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.
First, we observe that
Hatch quality is higher when the NRS has lower temperature than SRS, so we can assume . Then:
Since is a parabola pointing down, the sole CP is its absolute maximum. It's in the correct range, so and optimize combined hatch quality given the constraints.