MATH102 December 2019
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Question 04 (d)

Remember our fishy friends? The quality of hatching of larvae of coldwater 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.
Based on the temperature, we can predict how much energy a fish invests in finding enough food to produce healthy eggs, $E_{\mathrm {eggs} }$.
Fish also need to spend some energy in being alert for predators that want to eat them, $E_{\mathrm {alert} }$.
The fish try to minimize the total amount of energy used, $E_{\text{eggs}}+E_{\text{alert}}$, by seeking out areas of optimal temperature.
Using the graph above, choose the one statement below that best describes the situation.
 At lower temperatures, there are more predators, so the fish try to seek out waters that are warmer than $4^{\circ }$, but still cold enough to lay eggs in.
 The total amount of energy expended is optimal at around $7^{\circ }$, so the fish seek out waters with this temperature to find the most food to gain energy to lay their eggs.
 The warmer the water, the better for the eggs, because at high temperatures, the total amount of energy does not change much as temperature changes.
 Since there always will be predators around, the fish should focus on foraging to produce eggs, and stay in waters with the optimal temperature $3^{\circ }$.
 The total amount of energy invested is minimal in waters with temperatures of around $2^{\circ }$, although this temperature does not have the most abundant food.
 The energy spent towards being alert is highest at $5^{\circ }$ while the energy for producing eggs is lowest around $3^{\circ }$, so fish should thrive best in waters of around $4^{\circ }$.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Less energy being expended is better for the fish!

Checking a solution serves two purposes: helping you if, after having used the hint, 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.
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Solution

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The correct answer is E. The main error in each of the other choices is indicated below:
 At lower temperatures, there are more predators, so the fish try to seek out waters that are warmer than 4°, but still cold enough to lay eggs in.
 The total amount of energy expended is optimal at around 7°, so the fish seek out waters with this temperature to find the most food to gain energy to lay their eggs.
 The warmer the water, the better for the eggs, because at high temperatures, the total amount of energy does not change much as temperature changes.
 Since there always will be predators around, the fish should focus on foraging to produce eggs, and stay in waters with the optimal temperature 3°.
 The total amount of energy invested is minimal in waters with temperatures of around 2°, although this temperature does not have the most abundant food.
 The energy spent towards being alert is highest at 5° while the energy for producing eggs is lowest around 3°, so fish should thrive best in waters of around 4°.
