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Problem 47 Hard Difficulty

You are given a table of average monthly temperatures and a scatter plot based on the data. Use the methods of Example 8 to find a periodic function of the form $y=A \sin (B t-C)+D$ whose values approximate the monthly temperatures. The data in these exercises, as well as in Exercise $51,$ are from the Global Historical Climatology Network, and can be accessed on the Internet through the following website created by Robert Hoare.
(GRAPH CANT COPY)
Average Monthly Temperatures for Bangor, Maine
$$\begin{array}{lc}\hline \text { Month } & \text { Temperature ('F) } \\\hline \text { Jan. } & 18.0 \\
\text { Feb. } & 20.8 \\\text { Mar. } & 30.7 \\\text { Apr. } & 42.6 \\\text { May } & 54.1 \\
\text { June } & 62.8 \\\text { July } & 68.5 \\\text { Aug. } & 67.3 \\\text { Sept. } & 58.1 \\
\text { Oct. } & 47.1 \\\text { Nov. } & 36.9 \\\text { Dec. } & 23.7 \\\hline\end{array}$$

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Video Transcript

so to start solving this. The first thing we notice is that the maximum temperature occurs in July and that's maximum temperature is 68.5 and July is the seventh month and then the minimum temperature occurs in January, which is the first month, and the minimum temperature is 18 points years. So this is information that you can get from the till. Now. Let's calculate a, which would be cool to maximum minus two minima divided by two. So we have 68.5, minus 18.0, divided by two, and that's equal to 25.25 The crease pattern that next day since it's a year. So the petered is 12 and B is equal to two pi divided by the period, so that would be too high, divided by 12 which is power six. Now the main temperature, which is equal to D, will be equal to maximum temperature, plus the minimum temperature divided by two. So that would be 68.5 plus 18.0 divided by two, and that is equal to 43.25 So we've got mean temperature, which would give value for D now again, looking into the table. What? The finest at the mean temperature first occurs in April, which is teas equal to four so we can rank our equation. Why is equal to a 25.25 sign off pi over six times t by his four class 43.25? Or why is he put to 25.25 Sign or hi over 60 minus two pi over three plus 43.25

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