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The hours of daylight, throughout the year, in a particular town can be graphed using trigonometric functions. On June 21, the longest day of the year, there are 15.3 daylight hours. On Dec. 21, the shortest day of the year, there are 8.7 daylight hours.

 

a) Determine the function for the number of daylight hours with respect to the number of days since Jan. 1st.     

(Hint: Jan. 1st is day 1)

 

b) Determine the number of daylight hours the town will have on March 27th and October 2nd.

Guest Dec 10, 2017
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This will  not be exact  and I'm using degrees 

 

Because we have 365 days......then each  day will be 360 / 365  =  (72/73)°......thus, the funkiness of the measurements....!!!!

 

I'm using the sine function, here.....

 

The midline of this function will be    [15.3 + 8.7] / 2  =  24/2  = 12

 

The amplitude  is  3.3....and the graph will be shifted up 12 units

 

I'm letting  x = 0   =   Jan 1st......this is arbitrary......it won't make any difference as long as we shift the day count back by 1 unit......in other words....Jan 2nd  will be x = 1  on our graph instead of x = 2.......even so.....the true day count and days after Jan 1st will be "off" a little

 

So.......Dec 21st  is 11 days before Jan 1st

 

Because the  normal minimum for the sine  = -90°  we want to let the minimum occur at x = -11

 

So....we are shifting the graph  by   90  +  (-11)(72/73)  ≈ 79.1056° to the right

 

 

So......the approximate graph is

 

y = 3.3sin ( (72x)/73 -  79.1056) + 12

 

March 27th is the 86th day of the year - x ≈ 85 on the graph since Jan 1st  = 0 ......the approximate number of daylight hours on this day will be ≈  12.27 hrs   ....this makes sense....on March 21st there will be 12 hours and the days grow longer after this until June 21st

 

And October 2nd  is the 275th day of the year - x ≈  274 on the graph........and the approximate daylight hours at that time will be ≈ 11.365 hrs.....this also makes sense......the days grow shorter after Sept 21st (when there are 12 hours of daylight)  until Dec 21st

 

Here's the graph :https://www.desmos.com/calculator/wmtc8h5fss

 

cool cool cool

CPhill  Dec 10, 2017
edited by CPhill  Dec 11, 2017
edited by CPhill  Dec 11, 2017
edited by CPhill  Dec 11, 2017

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