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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:::info
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)According to the graph, on July 19 (day 200) there would be 14.24 hours.

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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) According to the www.timeanddate.com the actual number of hours of daylight for July 19, 2019 was 14.18 hours or 850.8 minutes. Therefore the model's prediction minus the actual = $854.4-850.8 = 3.6 minutes$. The model's prediction was off by 3.6 minutes.

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(4) Compute $D'(x)$. Show all work.
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(4)$D\left(x\right)\ =\ 12.1\ -\ 2.4\cos\left(\frac{2\pi\left(x+10\right)}{365}\right)$

$D(x) = 12.1-2.4\cos\left(\frac{2\pi x}{365}+\frac{20\pi}{365}\right)$

$D'(x)=2.4\sin\left(\frac{2\pi x}{365}+\frac{20\pi}{365}\right)\cdot d/dx\left[\frac{2\pi x}{365}+\frac{20\pi}{365}\right]$

$D'(x)=2.4\sin\left(\frac{2\pi x}{365}+\frac{20\pi}{365}\right)\cdot\frac{2\pi}{365}$

$D'(x)=\frac{4.8\pi}{365}\sin\left(\frac{2\pi x}{365}+\frac{20\pi}{365}\right)$

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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)
The rate of for July 19 (200 days) is at $-0.0188353725245$ hours per day. As time moves to 200 days, the rate of change is at −0.000313922875408 minutes per day.

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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) The day of the year that will be longest would be on day 172 or July 21, 2019.

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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) If we know the slope is increasing. We know that the derivative is positive or increasing. In addition we know days are increasing. Therefore we could find the D''(x) to tell us when the number of hours of daylight is increasing most rapidly.

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