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/dvu69qrvhr?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

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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) $D(200)=14.2360661834$ 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 site, there will be 14 hours 17 minutes of daylight or 14.283333333333 hours
The model's prediction is off by 0.047267149933 hours.
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(4) Compute $D'(x)$. Show all work.
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$D'\left(x\right)=\frac{d}{dx}12.1-\frac{d}{dx}2.4\cos\left(\frac{2\pi\left(x+10\right)}{365}\right)$
$D'\left(x\right)=0+2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\cdot\frac{d}{dx}\left(\frac{2\pi\left(x+10\right)}{365}\right)$
$=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\cdot\frac{2\pi}{365}$
(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)$D'\left(200\right)=2.4\sin\left(\frac{2\pi\left(200+10\right)}{365}\right)\cdot\frac{2\pi}{365}$
$=−1.09417606442$ hours/day
-60.0015 minutes/day
On July 19th the rate at which the number of hours of daylight is changing at a negative rate of change.

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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)$2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)=0$
<iframe src="https://www.desmos.com/calculator/rveb9na6es?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>
When the derivative is set equal to zero the x value or day is 172 when the day is the longest.

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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 find the day of the year when the number of hours of daylight is increasing most rapidly, you find where the derivative or slope is the greatest. You can use the central difference formula to find the central value of the slope when it is increasing.

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