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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)


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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) On July 19th, the 200th day of the year, there was 14 hours and 13 minutes 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 time and date website says that on July 19th, there was 14 hours and 17 minutes of daylight. Comparing that to our given model, which says July 19th had 14hr and 13 min, it looks like our model is under by about 4 minutes.

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(4) Compute $D'(x)$. Show all work.
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(4)

$D'\left(x\right)=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\frac{\left(d\right)}{dx}\left(\frac{2\pi\left(x+10\right)}{365}\right)$

$D'\left(x\right)=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\frac{\left(d\right)}{dx}\left(\frac{2\pi}{365}\left(x\right)+\frac{20\pi}{365}\right)$

$D'\left(x\right)=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\left(\frac{2\pi}{365}\right)$

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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)
$D'(200)= -0.1130118$ min/day

On July 19th (200th day), the rate at which daylight is decreasing is 0.1130118 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) Looking at the graph of D'(x), I noticed that the slope of the graph becomes 0 on the 172nd day, June 21st, which is the longest day of the year having 14 hours and 30 minutes of daylight.

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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 is increasing most rapidly, I could look at the graph and find when the slope is concave up and the steepest. I could also find when the number of hours is increasing most rapidly by finding the second derivative of D'(x).

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