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The table shows the average daily high temperatures in Houston $ H $ (in degrees Fahrenheit) for month $ t $, with $ t = 1 $ corresponding to January. (Source: National Climatic Data Center)

(a) Create a scatter plot of the data.

(b) Find a cosine model for the temperatures in Houston.

(c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data?

(d) What is the overall average daily high temperature in Houston?

(e) Use a graphing utility to describe the months during which the average daily high temperature is above $ 86^\circ $ and below $ 86^\circ F $.

(a) SEE GRAPH(b) $\mathrm{T}=77.95+15.65 \cos \frac{\pi}{6} t+\frac{5 \pi}{6}$(c) very well.(d) $6-9,10-5$(e) The average daily temperature is above 86 during the summer and below 86when it's not summer.

Precalculus

Algebra

Chapter 5

Analytic Trigonometry

Section 3

Solving Trigonometric Equations

Trigonometry

Introduction to Trigonometry

Oregon State University

Harvey Mudd College

Lectures

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so we need to make a scatter plot. So I went in my calculator, and I have a T 84 t 83 would act the same way. And you want to go in and plug in the time from one up to 12, and we find that the low temperature does seem to take place at that 62.3 degrees. Uh, the month of seven, which would be July, has that high of 93.6 and plug in all that data into the calculator, and we find that the scatter plot and I just hit Zoom, stay up and the scatter plot ends up doing something that looks, you know, the dads come along like this and they do something like that so it doesn't like a sign. Use little function. It's not exactly one, so let's analyze it. And so you could use a actually a trigger aggression and get the function. But the purpose of this question, I think, is for you to find the model yourself. So after you grab it on your calculator, let's find the model. So I know that I have a vertical shift and we need to find that middle of that grab. So if I average the 62.3 and the 93.6 Adam together divide by two, I could find out that the middle or the vertical ship is right here. That's this location. And then we can do either attraction. But we have to find what this. If we take the 77.95 which is in the middle, and find the difference between it and the high and low, we can find out what that length is, and that length comes out to be a 15.65 So we will say our amplitude is 15.65 degrees so we can start writing our equation Now. I also see that this will be at a low Hello at time one. It's also at a high at times seven. So depending on how you want a phase shift, you can phase shifted to either one. We also know the period it takes 12 months to finish a cycle. So we know that that coefficient on the angle will end up being My little B will equal to pi divided by the period. So we know that that coefficient needs to be pi over six. So I'm gonna read cosine function so we know why. Which will be the temperature is equal to, and I'm going to start mind down here, so I'm going to start it at a low. So I'm gonna put a The amplitude is 15.65 but I'm gonna put a negative on it and we have co sign of and our coefficient needs to be pi over six. Now, my low is taking place at pot at at one. And then we have our vertical shift at 77.95 degrees. So you want to distribute that through you may again, you can write a sine function and goto a middle, but this coastline function will fit quite well. And then you're asked to graph that. So if you take this function and put it into your graph or make sure you're in radiant mode and we'll find out that it does fit quite well, does fit quite well. And then he just asked kind of a simple question. What was the high in 93.6 is the highest average high temperature over that whole time I was in July and then you're looking at, when will you end up having the temperature be over 86? So you can actually with your calculator, you may wanna turn off your plot so there's not a conflict. Turn the stat plot off and you'll have that model. That's graft, and we'll find out that when basically and during the summer months, we have the average high that is higher than 86 degrees. And during the other months we have the average high lower than 86 degrees. And if you need to find those, exactly, you can use that intersect feature, or you could trace along the graph to find what this is and what this point is.

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