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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
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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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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/m3QWDDZ.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) ![](https://i.imgur.com/KCjY66F.png)



:::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) what we did was we imputted the given data using 14.236 hours of sunlight that was shown in the graph and then the given imformation which was being 14 hours 16 mins and 53 seconds and can be shown as 14.2813888 and in better terms can be shown as 14.2814 and we got that buy using a formula and set it up like this: 14.2814-14.236= 0.0454 hrs. 

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(4) Compute $D'(x)$. Show all work.
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(4) How this can be shown is like this (also we need to simplify the $\frac{2\pi}{365}=0.01721$) and that would be our number we would put in like so: 
$D'\left(x\right)=\left(\frac{d}{dx}\right)\left(12.1-2.4\cos\left(0.01721\left(x+10\right)\right)\right)$
$=0\left(2.4\right)\left(\frac{d}{dx}\right)\cos\left(0.01721\left(x+10\right)\right)$
now we apply the chain rule and that is set up like this formula: $\frac{df\left(u\right)}{dx}=\frac{df}{dx}+\frac{du}{dx}$ then we subsitute our numbers in that we know and use the formula above and it would like like this:
$D'\left(x\right)=-2.4\left(-\sin u\right)\left(0.01721\right)$ and plugging in the other information the final equation would look like this: 
$D'\left(x\right)=0.06121\sin\left(0.01721\left(x+10\right)\right)$

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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) The rate at which the number of hours of daylight chnaging on July 19th would = D'(200) since July 19th is the 200th day in the year. Then we can plug it into the equation that was used in question 4 like this: 
$D'\left(x\right)=0.04131\sin\left(0.1721\left(210\right)\right)$ and its 210 at the end because of the x+10 it was 200+10 which equals 210 then that equation equals:$0.041\left(-0.455\right)=-0.0188$. Now we have to multiply by 60 because that is how many hours our in a day. So the minutes that were shown in the equation was -1.128 minutes per day so that but not such thing as a negative minute so we just ignore it, and we find out that on July 19th the number of hours of daylight are changing is 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) $D'\left(x\right)=0$
$=\frac{4.8\pi}{365}\ \sin\left(\frac{2\pi\left(x+10\right)}{365}\right)=0$
$=\sin\ \left(\frac{2\pi\left(x+10\right)}{365}\right)=0$
$=\frac{2\pi\left(x+10\right)}{365}=n\pi$
$x=\frac{365n\pi-20\pi}{2\pi}$
$x=\frac{365n-20}{2},\ n=0,\ 1,\ 2,\ 3,\ ...$
$n=1\ \&\ x=172.5$
And the 172.5 day would be 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) D'(x) is maximum when D''(x)=0 and that means when the function changes the concavity from concave up to concave down as D'(x) is increasing and before that D''(x)=0 which that is the inflection point. Also to look at the increasing slopes and figure out which one would be the steapist or to find the largest tangent for that increasing section. 


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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.



