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)

:::info
(2) According to this model how many hours of daylight will there be on July 19 (day 200)?
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(2)
On the 200th day, there will be 14.236 hours of daylight according to this model.

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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 timeanddate.com there is 12.07 hours in a day so we are off by 2.17 hours.

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(4) Compute $D'(x)$. Show all work.
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(4)
$d(x)=12.1-2.4cos((2pi(x+10))/365$
$d'(x)=({24pi*sin({2pi*x+20pi}{/365})}{/1825}$

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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)
$d'(x)=({24pi*sin({2pi*x+20pi}{/365})}{/1825}$
$d'(200)=({24pi*sin({2pi*(200)+20pi}{/365})}{/1825}$
$=2.1707*10^{-4}$ MInutes/day is the rate at which the hours of daylight is changing at the point where x=200 days.
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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)
$d'(x)=({24pi*sin({2pi*x+20pi}{/365})}{/1825}$
$d'(0)=({24pi*sin({2pi*0+20pi}{/365})}{/1825}$
$=1.0343*10^{-5}$ minutes per day
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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 are approaching the longest day of the year when daylight is is increasing most rapidly, so we can assume that once we beging to decrease most rapidly the point inbetween these two direction should equal 0, where we are neither increasing or decreasing in daylight, so this should be our maximum point and the point at which $x$ is equal to our longest day of the year.

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