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)$D(200)=14.236$ 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)The model's prediction is off by .066 minutes from the actual minutes of 14.17 hours of daylight.

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(4) Compute $D'(x)$. Show all work.
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(4)$D(x)=12.1-2.4cos(\frac{2\pi(x+10)}{365})$
$=12.1-2.4cos((\frac{2\pi}{365})+(\frac{20\pi}{365}))$
$=0+2.4sin((\frac{2\pi\cdot x}{365})+(\frac{20\pi}{365}))$
$D'(x)=(\frac{4.8\pi}{365})sin((\frac{2\pi\cdot x}{365})+(\frac{20\pi}{365}))$

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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 at which the number of hours of daylight are changing on July 19 is -1.129min/day. This means that the hours of daylight are decreasing at this rate.

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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)$D'(x)=(\frac{4.8\pi}{365})sin(\frac{2\pi\cdot x}{365})+(\frac{20\pi}{365})$
$0=(\frac{4.8\pi}{365})sin(\frac{2\pi\cdot x}{365})+(\frac{20\pi}{365})$
$0=sin(\frac{2\pi\cdot x}{365})+(\frac{20\pi}{365})$
$\pi=(\frac{2\pi\cdot x}{365})+(\frac{20\pi}{365})$
$365\pi={2\pi\cdot x}+20\pi$
$365=2x+20$
$345=2x$
$172.5=x$
Summer solstice is on June 20, 2020 of this 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)To find when the number of hours are increasing most rapidly, you would find its derivative and set the derivative to equal zero. From an increasing to a decreasing state. I would set the second derivative to equal zero and solve equation. D''(x)=0. Solving this would give the maximum value of hours of daylight that are increasing the most.

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