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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)

$$y= -19.8\cos\big(\frac{\pi}{6}t-\frac{\pi}{6}\big)+85.3\\

\text{or}\\

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

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Numerade Educator

Campbell University

Oregon State University

McMaster University

We have average temperature data for one full year in Phoenix, Arizona, and we plotted this these points over a period of two years. So using this graph that we plotted these points onto, it appears that we have something of a periodic negative co sign curve going on here. That's gone over the course of two periods, each of which has a has a period of about 12. So we're going to write out a periodic curve in the forum Negative eight times co sign of BT, minus C plus de in something that will approximate this curve. So for now, I'm going to remove this curve so that this graph so that we can start finding what ABC and deer are going to look like. So let's start out with a so a corresponds with the amplitude. So in your textbook you found that a equals the maximum minus, the minimum divided by two. So that's going to be the maximum. Here is 105.1, minus the minimum, which is 65.5, divided by two. So that's going to be about 39.6, divided by two, which is equal to 19.8, So a is equal to 19.8. Next be so be it corresponds with the period. So the period is equal to 12 in this case because we're going over the course. 12345 689 10 11 12 months. So our period is 12. That's why we repeated it twice on our plot. So that's going to be equal to two. Pi divided by B 12 b is equal to two pi, so B is equal to two pi divided by 12 or pi over six. The next we're going to find to see. So for C, we're going to look at where the maximum would normally be. So in a period of 12 the maximum would usually be in the very center of the period for a negative co sign curve, so that maximum would be 1/2 the period, so it would be at six. Our maximum, as we already found, is that July, which is seven. So ours is at seven, which has shifted to the right by one. So C is going to be equal toe one to cause that shift by one. So that's right at what we have so far, so we have negative. A is 19.8 times co sign of B is pi over six times t minus. See is one plus de people's Why so now we just need to find what D is equal to here. Let's write D in purple so we're trying to find what day is equal to. So let's plug in, say, the first set up here So we'll plug in one for tea, so have negative 19.8 times Co sign of pi over six times one minus pi over six plus d equals 65.5. So we're going to have negative 19 times. Co sign of pie sixth minus pi of 60 Co Sign of Zero is once will have negative 19.8 times one plus key people 65.5 So adding negative 19. Adding 19.8 to both sides will get d equals 85.3. So our final curve is equal to negative. 19.8 times. Co sign of pi over 60 minus pi over six plus 85.3. I'm going to scroll down a bit now that we have this information. And I plotted this graph over the dots that we saw over earlier over two periods, and we're going to see that this graph actually matches fairly well with these temperature plots that we already had. So this seems to be a good fit for our plot.