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)
<iframe src="https://www.desmos.com/calculator/qphn6pqa01?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

:::info
(2) According to this model how many hours of daylight will there be on July 19 (day 200)?
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(2) $D(200)=12.1-2.4\cos\left(\frac{2\pi\left(200+10\right)}{365}\right)$
$=14.236$ hours of daylight in LV
According to this model, there will be about 14 hours and 14 minutes of daylight on July 19.

:::info
(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) The website states the daylight length would be 14 hours and 16 minutes.
The model's prediction is off by 2 minutes.

:::info
(4) Compute $D'(x)$. Show all work.
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(4)
$D(x)=12.1-2.4\cos\left(\frac{2\pi\left(x+10\right)}{365}\right)$
$D'(x)=\frac{d}{dt}\left[12.1\right]-\frac{d}{dt}\left[2.4\cos\left(\frac{2\pi\left(x+10\right)}{365}\right)\right]$
$=0+2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\cdot\left(\frac{2\pi\left(x+10\right)}{365}\right)$
$=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\cdot\left(\frac{2\pi x+20\pi}{365}\right)$
$=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\cdot\left(\frac{2\pi x}{365}\right)+\left(\frac{20\pi}{365}\right)$
$=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\left(\frac{2\pi }{365}\right)$

:::info
(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)
$D'(200)=2.4\sin\left(\frac{2\pi\left(200+10\right)}{365}\right)\left(\frac{2\pi}{365}\right)$
$=−0.0188$ minutes/day
The length of daylight decreases at a rate of 0.0188 minute a 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)
$0=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\left(\frac{2\pi}{365}\right)$
$0=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)$
$0=\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)$
$\pi=\frac{2\pi\left(x+10\right)}{365}$
$365\pi=2\pi\left(x+10\right)$
$\frac{365}{2}-10=x$
$x=172.5$ days
Day 172 will be the longest day of the year.
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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)
We can find the derivative of the derivative, so D''(x) in this case, because D''(x) depicts the change of the change in regards to the number of hours of daylight per day.

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