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.

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)\\[D(x)=12.1-2.4\cos \left(\frac{2\pi \left(200+10\right)}{365}\right)=14.236Daylight(h)/Day.
\\]

:::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)From the model's prediction and the actual number of minutes of dalight there is only a two minute difference. This was found by converting the approximation that was found of 14.236 Daylight(h)/Day and converting it to minutes which becomes 854.4 minutes then subracted it from the miuntes of the actual model which is 856.8 minutes. The difference then becomes 2.4 minutes.

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(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)=0+2.4\sin \left(\frac{2\pi \left(x+10\right)}{365}\right).\left(\frac{2\pi \left(x+10\right)}{365}\right)
\\]\\[
D'(x)=2.4\sin \left(\frac{2\pi \left(x+10\right)}{365}\right)\frac{2\pi}{365}
\\]

:::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)\frac{2\pi}{365}]=-.02\space hours/day
\\]
\\[(-.02\ hr /1 day )(60 min/1 hr)=-1.2 mins/day\\]
What this means is that on the day of July 19, the sunlight per day is decreasing by 1.2 minutes per day which correlates with the curve of the graph because you see on that day it begins to fall at that point.

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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 the longest would be 172 according to the graph.

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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 we find the second deriviative of the first derivative we found and use that to find where the slope increases the most.

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