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) 
]

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(1) This graph shows the formula of D(x) graphed. When D(200) is placed in desmos accounting for the days as D(x) represents days, then the answer to the equation is 14.2360661834 or approximately 14.24. This new number represents the hours of daylight on that day inputted, in this case it is day 200.

:::info
(2) According to this model how many hours of daylight will there be on July 19 (day 200)
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(2) Roughly 14.24 hours or 14 hours and 15 minutes of daylight hours should exist on day 200 or July 19th, 2020.

:::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) Upon visiting the site suggested, in the previous question, it is noted that the amount of daylight hours available in Las Vegas on day 200 on July 19th will be 14 hours and 17 minutes. It is noted that there is a difference of only 2.8 minutes from the calculations suggested time.

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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).
\\]
$$D'\left(x\right)=12.1-2.4\ \frac{d}{dx}\ \left(\cos\left(\frac{2\pi\left(x+10\right)}{365}\right)\right)$$

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

$$D'\left(x\right)=2.4\sin\left(\frac{2\pi\left(x+10\right)}{365}\right)\cdot\left(\frac{2\pi}{365}\right)$$
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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) By using the previous calculations of derivative and plugging in the 200 when D'(200), then we get −0.0188353725245 and we must multiply by 60 to account for the seconds to minutes and we get −1.13012235147 minutes of daylight per day.

:::info
(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 point where the slope is as close to 0 as possible you see that on day 172 or 173 it is the longest period of sunlight.

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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.

(7) Refering back to the graph all you do is look for the steepest positive slope.

you could just take the derivative and graph which will so the slope of the line and find the highest point which will suggest the steepest positive slope and indicate most increase.


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