the-table-shows-the-average-daily-high-temperatures-in-houston-h-in-degrees-fahrenheit-for-month-t-w

Question

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 $.

Sheryl Ezze · Accepted Answer

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&#x27;t like a sign. Use little function. It&#x27;s not exactly one, so let&#x27;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&#x27;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&#x27;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&#x27;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&#x27;m gonna read cosine function so we know why. Which will be the temperature is equal to, and I&#x27;m going to start mind down here, so I&#x27;m going to start it at a low. So I&#x27;m gonna put a The amplitude is 15.65 but I&#x27;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&#x27;re asked to graph that. So if you take this function and put it into your graph or make sure you&#x27;re in radiant mode and we&#x27;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&#x27;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&#x27;s not a conflict. Turn the stat plot off and you&#x27;ll have that model. That&#x27;s graft, and we&#x27;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.

Ankit Gupta · Answer

we have given the table. I&#x27;m showing you. So now here is the table. The table. So that normal daily high temperature in Honolulu, Hawaii, T in degrees Fahrenheit, Temperature related, flooded, hyped, and four months D. So here is the month B minus one. So, de with the venison corresponding to January, so now equals to one. Here, Jen. So this is the fare and so on. So visitor December And now we have to make a scatter plot. So for this here we have to make a graph. So this is the graph. And here, when they represent here. So this is the 1234 So one representative, 80.1. So this is the 0.80 point one. And then it did 80.2. So quite increasing and then 81.2. So this is here and that 82 point I&#x27;m here doing it for the four that is here, 82.7. So this is the 85 and 90. So this is 82 word seven and then 84 16 and then a diesel in and then 87.9 and then 88.6 services here. And then it this explains selling somewhere here and then 83.9. So this is here and then 81.2. So this is here. So this is the way Lucas pointing for the is get a lot. So this is like this. So this we have made. So now put this value and good a site sign on graphing calculator. Put this value to sign funds and then you&#x27;re graphing calculator, so Johnson will be off. Y equals two IHS side X.

Heather Zimmers · Answer

Okay, we&#x27;re using a graphing utility for this problem. So here&#x27;s my graphing calculator, and we start by going into the stat menu for statistics and then into edit. And you can see I&#x27;ve already typed in the numbers. So for list one, these air the month numbers where January is month 1 April is month for July, as 7 October is 10 and December is 12 and then enlist to I have the temperatures for New York City, an endless three a Have the temperatures for Fairbanks. Okay, so the first thing we&#x27;re going to do is find regression models or regression equations for each of these cities. So I press stat and then over to calculate. And then we&#x27;re going to look for a sign regression model. And for this calculator, its way down the list. You just keep scrolling down past the more familiar regression models, and finally we get to sign regression. So I&#x27;m going to select that, and I want to use list one and list to if I&#x27;m using New York City and and I&#x27;m going to leave some of these things blank or unchanged. But what I do want to do is store my regression equation in my Y equals menu. So where it says store regression equation, I&#x27;m going to press variables and then over toe why? Variables and shoes function and choose why one and then I&#x27;m going to calculate. So what you see here are all the numbers that you would round and write down to have your model. We have the values of A, B, C and D. And then if we look in the y equals menu Ah, you should see the equation pasted in there. And I&#x27;m just going to erase that second equation that was left over from some other time. Okay, Now we want to find the regression equation for Fairbanks. So once again, I pressed at and then over to calculate, and then we&#x27;re going to find sign regression again. And then this time I&#x27;m going to use list one and list three. So list one is for the month analysts threes for Fairbanks. So if I press second function three again l three and this one, I&#x27;m going to store as why, too. So I press variables over toe y variables, choose function, and then she was white, too. And then we calculate and these are all the values for that equation, so you would round them and write them down. And let&#x27;s look at the y equals menu. And there we have it pasted in as well. Okay, so we have our models. So now we move on to Part B, and we&#x27;re going to use these models to calculate the temperatures in different months. So I&#x27;m going to use the table for that. So first I&#x27;m going to go into my table set menu, which is second window. Make sure that it says independent ask, and that way you can type in whatever X values you want. Okay. From here, we&#x27;re going into the table menu, which is second graf and we want February, which would be month to So we just type of to in the UAE. One column. We see the New York City temperature, and in the Y two column, we see the fair beings temperature and then march would be month three and in May would be month five and June would be six. Angela and August would be 8 September would be nine and November would be 11. So now we have all the months covered okay for Part C, it says, Compare the models, and I think it would be nice to look at the graphs in order to do that. And so let&#x27;s go ahead and choose some window dimensions for these graphs. So press window. Remember, the X value represents the month, so we could go from zero to 13 and we would see a little bit more than a year. And then the Y value represents the temperature, so it gets pretty cold in Fairbanks. Let&#x27;s go down to negative 20 and then it got up to 70 something in New York. So maybe we go up to 85 and perhaps we go by tens and let&#x27;s take a look at that. So the first craft is showing us New York City&#x27;s temperature over the course of a year, and then the next graph is going to show us Fairbanks. So some comparisons that we could make our that on any given day New York&#x27;s temperature is going to be warmer than Fairbanks temperature, But we could also say that there&#x27;s a greater spread vertically for Fairbanks. We call that the amplitude, and so there&#x27;s more of a wide range of temperatures in Fairbanks, and then you could make other varied conclusions as well

Ava St. Clair · Answer

All right, this question. We&#x27;re gonna be modeling some signing. So functions were asking too. We&#x27;re being asked to model the temperature T as a Sinus. Total function of time. Using 36 is the minimum, and 79 is the maximum. Never going to grab our function with these guys What you&#x27;re given hear. So first, really, for the amplitude, the amplitude is going to be equal to half of the vertical distance during the minimums and the maximums. We&#x27;re gonna calculate this with our minimum value of 36 our maximum you 79 Jake 79 minus 36. We divide that by two Megan amplitude of 21.5. So let&#x27;s start drawing our son. You settle function. We&#x27;re doing it as a mop. Is a model of the temperature capital T as the function of time. Little tea, sir. Right. This is a big T little city and write in our amplitude of 21 point size and I&#x27;m going to use co sign for the Maya graph because co sign starts at a maximum at X equals zero. Next, we&#x27;re going to look at our phase shift. That&#x27;s what comes here. So I&#x27;m saying that I need to obtain a maximum at seven x at X equal seven because you see that it gets larger, it gets to seven, and then it starts decreasing. So using my ex value of seven, I&#x27;m going to place that in my face shift. And lastly, we need to calculate our vertical translation s so you can tell that this is not going to zero. So we know it&#x27;s going to be. We&#x27;re adding something and moving our graph up, and we&#x27;re going to catch it or face shift the opposite way. They calculate amplitude or simply way honestly, we take a R two minimums and let me take a minimum and maximum our maximum of 79 in our minimum of 36. We add them together. Do you find the half way point between them? We divided by two, and that would get us a value of 57.5 as our vertical translation up. I have run a fruit so we&#x27;re adding 57.5 to our sign away. Our coastline live, and that is our final answer. That is our function of temperature as a function of tea, Sinus motor function and just check. I have made a scatter plot with this equation. I mean, see that it lines up pretty well.

Robert Daugherty · Answer

Okay, section 1.6. So in section 1.6, we&#x27;re looking at problem number 24. Okay, we&#x27;re doing some trick modeling here. So we are, after, ultimately a function that looks like this. Why is equal to a sign of be T minus h plus K. And we&#x27;re looking at modeling the data that we see here. Okay, so we&#x27;ve got time in months. Okay, so we were saying that. Okay, this represents one complete cycle. One year of the temperature&#x27;s okay, so find the value of be assuming that the period is 12 months, so we know that be is equal to two pi over the period. So in this case, that&#x27;s gonna be two pie, and they tell us that is 12 months, So that is going to be pi over six. So in this equation here, I already know that b is going to be pi over six. Okay, now, the next thing I needed to figure out the amplitude. So to figure out the amplitude, you know what happens with a sine function. So a sine function does something like this. So the amplitude is going to be this distance divided by two Okay. So if I look and see what are the maximum values? It looks like a T is the peak value and 30 is the minimum value. So it looks like my amplitude is going to be a T minus 30 over to which is 25. So I now know what the amplitude of this function is going to be. Okay. And if I know that that&#x27;s my amplitude. So, for example, if I know that this is my low point and this is my high point than 25 in between there. So I know that, kay, that vertical distance there is going to be at 80 minus 25 or 30 plus 25 so I can see the K, which is my vertical shift is going to be equal to 30 plus 25. So that is 55. So the only thing left to figure out now would be a TSH. Okay, so I got to figure out okay. At what spot? Um, would this graph, um, be shifted? Okay, So what I want to do is just take a look and see. Um, I&#x27;ve got a look and see at at that point. Uh, what would give me a value of 55? Okay, so let&#x27;s figure out that out in just a moment. So in this case, if my vertical shift is gonna be at 55 I&#x27;m looking at a sine function, a sine function starts zero. Where&#x27;s the closest I can get to 55? That list is gonna be a time equal five months. So I&#x27;m going to set a CI equal to five. And that&#x27;s where I&#x27;m going to come up with the final Quite final graph for this is gonna be why is equal to what is our amplitude 25 sign. What is B? B is pi over six. So it&#x27;s pi over six times and then we&#x27;re gonna have t minus five plus 55. Okay, so that&#x27;s the rough estimate of this grab. Based on the data, I would need to input into regression to something a little bit more accurate than what you see there. But real quick. Let&#x27;s just look and see what this looks like on a graphing calculator. So you see the points. So now I&#x27;ve graphed the points from 1 to 12 the different temperatures, and then if I look at the function that I just came up with. Ah, you can see the function graft here. So again you see a function that is graft. It looks like a sine function. And so we said that the vertical shift was at 55. So if the vertical shift is at 55 then you see this sign function. You see where it starts? One cycle at about five months. That&#x27;s where we see that the normal sign curve starting eso you see the shift of that, okay.

Charles Carter · Answer

Okay, so partly on this, just ask sister. You&#x27;re all quick sketch of, ah, scatter plot. Okay, so it starts at 45 goes down a little bit 44 and then it starts climbing up again. 63. That it repeats 63 then back down the 59 54. Okay. So because this has, uh, looks like it&#x27;s gonna have a turning point. How to turning points? I would say this looks like a cubic a regression would be the best. Fifth. Okay. Apart. Being just wants me to find that the average rate of change So we need to find, uh, amount of time that changes is three hours, and then we need to find the difference in temperature. So 9 a.m. to noon. Gonna be 51.1 to 57.97 to 57.9. My mr 21.1. Article 6.8. Find about three. Uh, which is gonna be around 2.26 degrees per hour. Good. Increasing. Okay. See, I would see average rate from three PM to six PM so we&#x27;re still dividing about three deserving a three hour change. Uh, three PM would be 15. 16 would be 18 63 minus 63. So the average rate of change of just be zero degrees per hour for those three hours insiders deem, Um, I think I said cubit. And then I made a cubit regression model and then use the function to predict the temperature at five. Okay, obtaining that function plugging in five will give us Ah, we&#x27;re looking in five to the function. Give me a temperature of around 45.2 degrees it&#x27;s in, and then plugging this into our equation and plotting this cubic function it&#x27;s gonna give us it starts beneath and goes back up through this, too, because of the understanding that over that particular one and something somewhere to that. So e that&#x27;s our green line that we just drew. Um, okay, and then after says, interpret the line intercept. And so the whiner stepped would just be the temperature, uh, midnight, because that would be when the time is equal to 00 hours after midnight is equal to midnight. Thank you very much.

Problem

The area of a rectangle (see figure)inscrib…

Problem 96 Easy Difficulty

Answer

(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 86
when it's not summer.

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(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 with Limits

Grace H.

Heather Z.

Kayleah T.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

Ron LarsonPrecalculus with Limits

Grace H.

Heather Z.

Kayleah T.

Maria G.

(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 86
when it's not summer.

Ron Larson

Precalculus with Limits