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/xgmvmlqxzh?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) <iframe src="https://www.desmos.com/calculator/3nkfrhcicf?embed" width="500px" height="500px" style="border: 1px solid #ccc" frameborder=0></iframe>

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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 source, there were 14.17 hours of daylight on July 19th of this year. In the model's prediction, there were 14.236 hours in this day. Now
$4.236-14.17=0.066$ so the prediction was off by $0.066$ hours or about $4$ minutes.

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(4) Compute $D'(x)$. Show all work.
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(4) We need to start by simplifying the expression $\frac{2\pi}{365}=0.01721$ and we would do that by substituting it in as so:
$$D'(x)=(\frac{d}{dx})(12.1-2.4cos(0.01721(x+10)))$$
$$=0(2.4)(\frac{d}{dx})cos(0.01721(x+10))$$
Here we would use the Chain Rule and substitute in our given values:
$$D'(x)=-2.4(-sinu)(0.1721)$$
$$D'(x)=0.06121sin(0.01721(x+10))$$

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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 19th is $D'(200)$ since that is the 200th day of the year. Since $200$ became our $x$ value, we need to substitute it into our equation:
$D'(200)=0.0413sin(0.1721(200+10))$
$0.041(-0.455)=-0.0188$ > now this equation needs to be mutiplied by 24 to represent the minutes in an hour and from this we get $-1.128$, negative minutes don't exist SO we would just say that the numbers of hours of daylight were changing at 1,128 minutes per 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) $D'(x)=0$
$=\frac{4.8\pi}{365}sin(\frac{2\pi\left(x+10\right)}{365})=0$
$=sin(\frac{2\pi\left(x+10\right)}{365})=0$
$=\frac{2\pi\left(x+10\right)}{365}=n\pi$
$=\frac{\left(365n\pi-20\pi\right)}{2\pi}$
$=\frac{\left(365n-20\right)}{2}, n=0,1,2,3...$
$n=1\&x$ $=172.5$
The 172.5 day of the year would land right on June 21st.

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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) You can see what the day with the longest number of daylight hours is by seeing where the amount of hours would start concaving down (as that indicates you're at the maximum and going back to shorter days). You could also measure separate slopes and determine the steepest or largest tangent where the graph should be increasing.

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