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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 Cape Town, South Africa$$\begin{array}{lc}\hline \text { Month } & \text { Temperature }\left(^{\circ} \mathrm{F}\right) \\\hline \text { Jan. } & 69.8 \\\text { Feb. } & 70.0 \\\text { Mar. } & 67.8 \\\text { Apr. } & 63.1 \\\text { May } & 58.8 \\\text { June } & 55.6 \\\text { July } & 54.3 \\\text { Aug. } & 55.4 \\\text { Sept. } & 57.7 \\\text { Oct. } & 61.2 \\\text { Nov. } & 64.8 \\\text { Dec. } & 67.8 \\\hline\end{array}$$

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##### Kristen K.

University of Michigan - Ann Arbor

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So first thing we notice is that the maximum temperature appears in February, which is the second month, and that maximum temperature is 70 points. You and the minimum temperature happens in July, which is the seventh month, and that temperature is 54.3 degrees. So we can calculate the amplitude A, which is equal to the maximum minus, the minimum divided by two. So we have 70 once you're minus 54.3, divided by two, and that equals 7.853 start night. Next for a year. The Peter is 12 months, so therefore be which is equal to two kind of are. But a period would be to pie divided by 12 and that's equal to tire work. Six. Now the next thing we can do is we can calculate the mean temperature, which would be equal to D, and that's equal to the maximum temperature plus the minimum temperature divided by two. So we got 17 points year old plus 54.3 divided by two, so we get D's equal to 62.15 decrease Fahrenheit now. The mean temperature first occurs in April, for which he would be equal before and plus in April. What we find is that going forward from April, the temperature will decrease. So therefore the martyr were being. Why is equal to negative? A, which is 7.85 Sign high over six times T minus four and plus 62.15 Or why is equal to negative 7.85 Sign high over six times T minus two pi over three plus 62.15

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##### Kristen K.

University of Michigan - Ann Arbor