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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: http://www.worldclimate.com/.

(TABLE CANNOT COPY)

$$25.25\sin\big(\frac{\pi}{6}t-\frac{2\pi}{3}\big)+43.25$$

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Campbell University

Harvey Mudd College

Baylor University

Idaho State University

we have here a table and a plot for the temperatures in Bangor, Maine, over one year. We plotted this over two years in order to have more than one full period off this graft. So let's try to write a periodic equation in the form a time sign of Bt, minus C plus D that will fit onto this curve. So for now, we're going to remove this curve and will come back with these plots and will come back to it later. So let's start out by finding a so it has to do with the amplitude. So to find a we're going to take the maximum minus the minimum, and divide that by two. So it's going to be our maximum. Looking at our graph, it seems that 60 year, our table, it seems that 68.5 is our maximum, and our minimum is going to be 18 0.0. Divide that by two, so we'll get 50.5 divided by two, which is equal to 25.25. So that's our A value. Next we're going to be finding be, which has to do with the period, so one period goes from January to December 123456789 10 11 12 months. So our period is 12. That are period is going to be equal to two ply divide if I'd be so that's going to be 12 times T equals two pi. In other words, be equals pi over six. So next we're going to be finding ever see value. So C is going to be found by looking at where the maximum of our table is so regularly. With a period of 12 our maximum will be at three, which is 1/4 of the way through since our sign graph usually looks like this with art maximum 1/4 of the way through the period. So in our case, we have our maximum at July which is at seven. So that's a shift of four. So that means that our C is going to be equal to four times our be a value which is four times pi over six. So this is going to be four pi over six or two pi over three. So now let's see what we have so far of this we have 25 points 25 time sign of pi over six times T minus two pi over three. Krusty equals y. So now we just need to find this devalue here, and we're going to do so by plugging in the data for January. We could choose any month, but January is just probably going to be the easiest, because that gives us a T A value of one. So that means we're going to have 25.25 times Sign of pi over six times one minus two. Pi over three plus B is equal to 18.0 is R y value here, so have 25.5. Sign of pi over six minus two pi thirds is gonna be a sign of pi over six minus four pi over six. That's a sign of negative three pi over six. Sign of negative high house, which is going to be equal to negative one. So this is 25.25 times negative. One plus D is equal to 18.0, so we can add 25.25 to the sides. So that will give us the equals 43.25 The our complete periodic equation is going to be 25.25 Sign time. Sign of pry over six times T minus two pi over three plus 43 recorders equals Why? So now let's scroll down now that we have this equation and see how it looks on our plot. So I plotted this on Dismas and wrote our friend Rupert's curve that we just found over it. And actually, this sign curve actually matches up pretty nicely with these plots of our temperature values. So this is going to be a pretty good approximation for these for the temperature in this city and