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title: 'Math 181 Miniproject 5: Hours of Daylight'

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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
===

**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)
![](https://i.imgur.com/MxjvFNI.png)




:::info
(2) According to this model how many hours of daylight will there be on July 19 (day 200)?
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(2)
According to the model on July 19 (day 200) there will be 14.236 hours of daylight. 


:::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)
14:16:53 = 14.28138
Differece = 14.28138 - 14.236 = 0.04538 hours = 2.7 minutes
The model's prediction is 2.7 minutes off.

:::info
(4) Compute $D'(x)$. Show all work.
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(4)
D(x)= 12.1 -2.4 cos ((2π(x+10))/365)
= 12.1 -2.4 cos (0.01721(x+10))           ---> chain rule = -2.4(-sinx)(0.01721) 

D'(x)= 0.04131 sin(0.01721(x+10)).


:::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)
July 19 = 200 days
D'(x)= 0.04131 sin(0.01721)(x+10)

= 0.04131 sin (0.01721)(200+10) = 0.04131 sin (0.01721)(210) = 0.041(-0.455) = -0.0188 hours/day = -1.128 min/days. 

:::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)D'(x)= 0
0.04131 sin(0.01721)(x+10)= 0
sin (0.01721 (x+10))=0
2π/365(x+10) = π = 172.5 
June 21st is probably the longest day


:::info
(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)


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To submit this assignment click on the Publish button ![Publish button icon](https://i.imgur.com/Qk7vi9V.png). Then copy the url of the final document and submit it in Canvas.



