MATH 130 Project
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School
University of North Carolina, Chapel Hill *
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Course
130
Subject
Mathematics
Date
Feb 20, 2024
Type
docx
Pages
12
Uploaded by MegaCheetahMaster93
MATH 130 Project
Modeling Data with Trigonometric Functions Alyssa Knott, Michaela Hallman, and Nava Little Planning a Trip to Mars
1)
The distance between Earth and Mars is constantly changing because both planets orbit the sun at different speeds, and neither planet has a perfectly circular orbit. Both have a somewhat elliptical orbit shape to their orbit, meaning they will not be locked in at a certain distance from one another at all times.
2)
It takes approximately 687 Earth days for Mars to complete a full orbit around the sun, and this is because Mars orbits at a much slower pace than Earth.
3)
The independent variable is the days, and the dependent variable is the distance between Mars and Earth. We decided to make the dependent variable the distance because the amount of distance between Mars and Earth is entirely dependent on however many days have passed since day 0. 4)
The best model fit for this data is a sinusoidal model. This is because the pattern of data repeats in a constant wave motion- as the number of days that passes increases (until a
certain point) so does the distance between Earth and Mars (until a certain point). This pattern is the same with the decreasing repeats of the data. Developing a Model for a Trip to Mars
5)
6)
1.72 (* 10
8
km) sin (
π
/386) + 2.27 (* 10
8
km)
7)
This function is relatively in-line with the graphed data. Some points do not fall exactly on the line, but they are very close and the line touches 5/6 of the data points.
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8)
9)
Above is the graph of the sinusoidal regression model (in red), the residuals (purple points), our calculated trigonometric model (in blue) and the actual data points (red points). The equation of the sinusoidal regression model is y = 2.41 + 1.69sin(.008x – 0.136) The regression model fits the data in a similar way that our data does, with an r
2
value of .9887, which means it fits approximately 99% of the data. Traveling to Mars 10) The most suitable day, from January 1, 2030, to send a human to Mars would be November 23, 2030. This is because we would want a human to land on Mars when it is at its closest distance to Earth, in order to ensure the shortest travel time possible. It takes humans approximately 2 years (or about 274 days) to travel to Mars. Considering Mars is
at its closest 600 days after day 0 (which in this case would be January 1, 2030), we would want to send humans 274 days before 600 days has lapsed. This means that the humans would land on Mars on August 24, 2031, which successfully reaches the goal of getting humans to Mars by the 2030’s.
Weather Fluctuations
11)
The independent variable is month of the year (for example, January = 1, May = 5, September = 9, etc.) and the dependent variable is the temperature. We made the months the independent variable because the temperature of a day depends on the time of year in which the data is being collected.
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Developing a Model for Temperatures
12)
13)
y = 19.5 (°F) cos (
π
6
– 6
π
) + 70.5 (°F)
14)
The model fits the data relatively well, but the phase shift is approximately .3 points off, therefore it is not as accurate as it should be. The phase shift we calculated cannot be changed as we used the information from the data the best we could.
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15)
Above is the sinusoidal regression model compared to the trigonometric model we calculated. The sinusoidal regression model is the line in green, and the model we calculated is in purple. The actual data points are in green as well. The equation for the regression model is y = 70.42 + 18.91sin(.52x – 2.1) The sinusoidal regression model fits the data better than our model does, but when comparing the equation of the regression model to ours, we can see that we were very close. 16) We don’t think the model we created would fit the data 100% correctly, but it would be relatively close. The model we calculated for the 2 years of average data was only .3 points off in the phase shift, so it might be slightly more inaccurate with less information such as 1 year of data. Nonetheless, the graph only off by a few decimal points, so it may not be significant when calculating temperature as both the temperature and monthly data
are not collected with anything beyond a decimal place.
Sources
Tillman, N. T., & Dobrijevic , D. (2022, January 20). How long does it take to get to Mars? Space.Com. https://www.space.com/24701-how-long-does-it-take-to-get-to-mars.html
Dobrijevic, D. (2022, February 4). Distance to Mars: How far away is the Red Planet? Space.Com. https://www.space.com/16875-how-far-away-is-mars.html
(2020, July 13). How Long is a Year on Other Planets? Nasa Science ; US Government . https://spaceplace.nasa.gov/years-on-other-planets/en/
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