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) The following is a link to the graph I created for this problem, using demos. Units are in hours of daylight for the xy-axis, and per day for the x-axis.


:::info
(2) According to this model how many hours of daylight will there be on July 19 (day 200)?
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(2) As I have shown by graphing in the link above, D(200) is equal to 14.2360661834. This means that there will be 14.2360661834 hours of daylight on day 200. I have linked a graph that shows the point on the graph represented by this number.


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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 website, the actual day length is 857.867 minutes on July 19th. The graph estimates this day will have 854.164 minutes of daylight. Thus, the model's prediction is off by 3.703 minutes.

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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)=12.1-2.4(d/dx)[cos((2pi(x+10)/365)])]
D'(x)=-2.4(-sin)((2pi(x+10)/365))*(2pi/365)
D'x=2.4sin((2pi(x+10)/365))*(2pi/365)

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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 following is the link to the graph I created in Desmos to illustrate this problem.

https://www.desmos.com/calculator/x8ahz0ihp2

I calculated this rate by using G(x) as an annotation so that Desmos would see the equation. After I plugged in 200, which is also the day of July 19, the answer given is 0.018. The answer was multiplied by 60 to get minutes per day. Thus, on July 19 the rate at which the hours of daylight are changing is - 1.130 minutes per day. The results here tell us that the number of hours of daylight is decreasing at a negative rate of 1.13 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) One trick to answer this question would be to actually zoom in on the graph of D'(x) where the daylight lasted the longest and the slope was zero. When you do that, you see that the point we are looking for lands on 14.5 hours on day 172. This answer is definitely realistic because day 172, or June 21, is the in the middle of the summer solstice and is also
the longest day of the year.

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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)The derivative of a function shows the slope. The slope is the rate of change of a function. If we graph the derivative of the function in Desmos, we can view the rate of change and locate where the hours of daylight is increasing the most. If we wanted to use the second derivative to give a more accurate answer, we could calculate that as well. The second derivative would show the rate of change of the actual slope.

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