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Problem 50 Hard Difficulty

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)
The following two scatter plots and table display average temperature data for two locations with very different climates: Dar es Salaam is south of the Equator, in Tanzania, on the Indian Ocean; Tiksi is in Russia, far north of the Arctic Circle, on the Arctic Ocean.
Average Monthly Temperatures for Dar es Salaam, Tanzania, and Tiksi, Russia
(TABLE CANT COPY)
(a) 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 for Dar es Salaam.
(b) Follow part (a) for Tiksi.
(c) By looking at the scatter plot for temperatures in Dar es Salaam, say if the average rate of change of temperature over the interval from January through July is positive or negative.
(d) Use the table of values to compute the average rate of change of temperature in Dar es Salaam over the interval from January through July. Be sure to include units as part of your answer, and check that the sign is consistent with your response in part (c).
(e) By looking at the scatter plot for temperatures in Tiksi, say if the average rate of change of temperature over the interval from January through July is positive or negative.
(f) Use the table of values to compute the average rate of change of temperature in Tiksi over the interval from January through July. As before, be sure to include units as part of your answer, and check that the sign is consistent with your response in part (e).

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Top Algebra Educators
Anna Marie V.

Campbell University

Alayna H.

McMaster University

MG
Maria G.

Numerade Educator

Michael J.

Idaho State University

Video Transcript

So for part A, we're looking at the information for terrorists alarm, and the first thing you notice is that the maximum temperature occurs in February, which is the second month, and that maximum temperature is 81.7. The minimum temperature occurs in July, which is the seventh month, and that minimum temperature is 73 point. Not now, using this week uncalculated value for in which is maximum minus the minimum divided by two. So that would be any 1.7 minus 73.9 divided by two, and that equals 3.9 now. The next thing is, pete is 12 months. So that means B, which is two pi divided by the period. So two part about it by 12 equals pi over six. So next week after let the mean temperature and the mean temperature would actually be the value 40 which would be diverted the shift. And that equals the maximum plus the minimum divided by two. So maximum temperature, plus the minimum temperature right about two. So we have 81.7 plus 73.9, divided by two. So we got a value for D, which is equal to 77.8. Now, again looking in the table, what we find is that the mean temperature. First, others in it's true. And for April T is equal to four and also in April. What we find is that the temperature is decreasing. So going from February to March to April 2, main a temptress decreasing, so the equation would have a negative value for a so this would be wise equal to negative 3.9. Sign off high over six times T minus four plus 77.8 or we air Why is equal to negative 3.9. Sign off pi over six T minus two pi over 30 and then plus 77.8. So that's nutrition that we have four Rossella. Now let's take a look at what is happening with the other City tells key. So it's talking with part B here for Till ski. Okay, so now, for thing is that the maximum temperature occurs in July, which is the seventh month, and that maximum temperature is 43.9 increase. The minimum temperature occurs in January, which is the first month, and that temperature is negative 24.2 degrees Fahrenheit. So from these two week and cal ability values the amplitude a, which is maximum minus the minima divided by two. So we got 43 59 minus negative, 24.2, divided by two, and that comes out to be equal to 34 point or five. Next, the PD again is 12 months, and that will be is equal to pi work. Six. Remember easy to fight over the period. Now the next thing is the mean temperature. So what is the mean temperature attack would give the value for key, which is also the vertical shift, and that's equal to look maximum plus the minimum divided. But so we have 43.9 plus negative 24.2 divided by two, and that comes out to people to 9.75 suggest a vertical shift, and then the mean temperature first occurs in in for for which TZ pulled before. So the equation would be why is equal to 34.45 sign off high over six times T minus four last 9.75? Or why is he quick oot 34.5 Sign off kite over 60 minus two pi over the plus 9.75 So that's your heart beat now for park si. What we see for our salam is that temperature decreases cause capture is DPC. Therefore, rate of change off temperature is negative. And for Parc de, we actually have to find that rate of change of temperature, so that would be equal to the temperature in July, which is 73.9 minus the temperature in January which is 81.3, and divide that by seven minus one. So that comes out to be too negative, 1.2 three beating and it would be degrees Fahrenheit per month. So that's the rate of change of temperature from January to July 4000 and now for part E asking whether TTF the rate of change of after his positive or negative Fortensky so again looking at the table and what we find is that the temperature increases and if the temperature is increasing, then this means that rate of change off temperature is positive and the last thing that we have to capital it is theatrical rate of temperature for till ski so raped off. Change off temperature for people to the temperature in July, which is 43.9 minus the temperature in January, which is negative, 24 point to divided by seven minus one, and that is equal to 11 point city five decrease Fahrenheit per month.

Other Schools
Top Algebra Educators
Anna Marie V.

Campbell University

Alayna H.

McMaster University

MG
Maria G.

Numerade Educator

Michael J.

Idaho State University