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

MATH102 December 2019

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Question 04 (b)

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. As we saw in OSH and WeBWorK, we can sometimes find a non-linear equation that best fits our data by changing the variables in the non-linear equation to make a linear equation, and then using the formula for line of best fit. Let $T>0$ be the temperature in Celsius, and let $Q$ be the hatch quality. You decide to fit a function $Q(T)=10+a(T-4)^{2}$ to the data in Columns A and B of the table below, choosing the parameter $a$ to make the equation fit the data as well as possible. In order to find the best value of $a$, you will use the data you have to compute appropriate values of $x$ and $y$, then use our formula for a line of best fit passing through the origin. This is done in the spreadsheet below. An arrow indicates the contents of a cell is copied down its column. The entries in Row 1 are labels. What should you write in cells C2, D2, E2, and F2?

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
Note that $x=(T-4)^{2}$ and $y=Q(T)-10$.

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. 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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. According to the table headers, we must have ${\color {blue}{\begin{aligned}{\texttt {C2}}&&&{\texttt {=(A2-4)}}{\hat {}}{\texttt {2}}\\{\texttt {D2}}&&&{\texttt {=B2-10}}\\{\texttt {E2}}&&&{\texttt {=C2*D2}}\\{\texttt {F2}}&&&{\texttt {=C2}}{\hat {}}{\texttt {2}}\end{aligned}}}$.