Math 181 Miniproject 5: Hours of Daylight.md
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tags: MATH 181
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Math 181 Miniproject 5: Hours of Daylight
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**Overview:** This miniproject will apply what you've learned about derivatives so far, especially the Chain Rule, to analyze the change the hours of daylight.

**Prerequisites:** The computational methods of Sections 2.1--2.5 of *Active Calculus*, especially Section 2.5 (The Chain Rule).

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The number of hours of daylight in Las Vegas on the $x$-th day of the year ($x=1$ for Jan 1) is given by the function together with a best fit curve from Desmos.}[^first]
[^first]: The model comes from some data at http://www.timeanddate.com/sun/usa/las-vegas?

\\[
D(x)=12.1-2.4\cos \left(\frac{2\pi \left(x+10\right)}{365}\right).
\\]
(1) Plot a graph of the function $D(x)$. Be sure to follow the guidelines for formatting graphs from the specifications page for miniprojects.
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(1)

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(2) According to this model how many hours of daylight will there be on July 19 (day 200)?
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(2)
Based upon this model, on the 200th day (July 19th) there is roughly 14.24 hours of daylight?

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(3) Go to http://www.timeanddate.com/sun/usa/las-vegas? and look up the actual number of hours of daylight for July 19 of this year. By how many minutes is the model's prediction off of the actual number of minutes of daylight?
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(3)
There are 14 hours and 18 minutes daylight on July 19 of this year according to the website provided. This total deviates from the previous question's estimate of 14.24 hours, making it a slight difference of 3.6 minutes.

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(4) Compute $D'(x)$. Show all work.
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(4)

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(5) Find the rate at which the number of hours of daylight are changing on July 19. Give your answer in minutes/day and interpret the results.
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(5)
d'(200) represents the rate of change for the daylight hours of July 19th, where there is loss of 1.13 minutes of daylight.

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(6) Note that near the center of the year the day will reach its maximum length when the slope of $D(x)$ is zero. Find the day of the year that will be longest by setting $D'(x)=0$ and solving.
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(6)
When setting d'(x) to equal 0, we can determine that the 172.5th day of the year (June 21st) is the longest day and has the most daylight hours.

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(7) Write an explanation of how you could find the day of the year when the number of hours of daylight is increasing most rapidly.
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(7)
To determine the day of the year in which daylight hours are increasing most rapidly, you would graph the daylight hours (hours per day) and set it as your x-axis and then set your y-axis to days of the year (365). From this you could then determine the longest day, find its point, and see where it has a slope of or close to 0. Once we have that done, we can visually see at which point of the graph is the line is rising the most rapidly.

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