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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 Phoenix, Arizona (based on daily maximums)

$$\begin{array}{lc}\hline \text { Month } & \text { Temperature ('F) } \\\hline \text { Jan. } & 65.5 \\

\text { Feb. } & 70.2 \\\text { Mar. } & 75.2 \\\text { Apr. } & 84.6 \\\text { May } & 93.2 \\\text { June } & 102.7 \\\text { July } & 105.1 \\\text { Aug. } & 103.1 \\\text { Sept. } & 98.6 \\\text { Oct. } & 88.2 \\

\text { Nov. } & 74.7 \\\text { Dec. } & 66.4 \\\hline\end{array}$$

$y=19.8 \sin \left(\frac{\pi}{6} t-\frac{2 \pi}{3}\right)+85.3$

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Idaho State University

so you can get the data. A couple of things come here. So four thing is that the maximum point occurs in the seventh month and the highest temperature is 105.1 and the minimum temperature occurs in January, which is the first month of the year, and that minimum temperature is 65.5 degrees Fahrenheit. So forth thing we can calculate is the amplitude, which is maximum temperature minus the minimum temperature divided by two. So that would be 100 and 5.1 minus 65 point 5/2. So we get a value for a which is equal to 1928 now. The next thing we notice, since it's a year, so the period is going to be 12. So be which is to pine divided by a period would be to pie divided by 12 which is higher work six. So we got a Reebok be now. The next thing that we can calculate is D, which is the vertical shift so D is equal to maximum, plus the minima divided by two. So we have 105.1 plus 65.5 and divide that by two. That gives you 85.3. Now. The next thing we need to do is where we first get this mean temperature. So this D would actually be the mean temperature, so the mean temperature first occurs. I didn't pay pro aan act Mies. That's when T is equal to four. So now we are ready to fight our equation. Which would be why is equal to a, which is 19.8 sign times pi over six. This is B Times T minus four and then plus D, which is 85.3 So expanding that Why is he going to 19 point it sign high over six times T minus two pi over three plus 85.3