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# trigonometry

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In New York City at the spring equinox there are 12 hours 8 minutes of daylight. The longest and the shortest days of the year vary by 2 hours 53 minutes from the equinox. In this year, the equinox falls on March 21. In this task, you’ll use a trigonometric function to model the hours of daylight hours on certain days of the year in New York City.

1.Find the amplitude and the period of the function.

2.Create a trigonometric function that describes the hours of sunlight for each day of the year.

Nov 4, 2018

#1
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The amplitude will be 1/2 of the variance in day length    2 hrs 53 min x 1/2 = 1 hr 26 1/2 min

The period is 365 days  (in a year)

Here is a graph (and the equation) with March 21st being the origin , x axis is days past March 21st and the y-axis is day length.

1.4416 is 1 hr 26 1/2 min in decimal    .9863 is the degree value of each day(360 degrees  in 365 days)

and 12.133   is the decimal equivalent of baseline equinox value of  12 hrs 8 miin

Note:   I am not sure what 'vary by 2 hr 53 min from equinox' exactly means.....if it means the shortest day is 2 53 shorter and the longer day is 2hr 53 longer (which I now think it might) then the amplitude is 2 hr 53 = 2.8833 hr    and  the equation becomes

2.8833 (sin .9863x) + 12.133    and is shown here:

Nov 4, 2018
edited by ElectricPavlov  Nov 4, 2018
edited by ElectricPavlov  Nov 4, 2018
edited by ElectricPavlov  Nov 4, 2018
#2
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can you please help with how to find out how many fewer daylight hours February 10 will have than March 21,

thank you

spirit1  Nov 4, 2018
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Yes,    slide backward from March 21st by 38 days to get to Feb 10th the read the 'y' value....or slide forward 328 days to Feb 10th,though you would really have to zoom in on the graph...

Use the equation     2.8833 sin (.9863(328)) + 12.133 = 10.42 hr = 10 hr 25 min

Here is a zoom-in of the graph:

ElectricPavlov  Nov 5, 2018
#3
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Modeling Ocean Tides

In one day, there are two high tides and two low tides in equally spaced intervals. The high tide is observed to be 6 feet above the average sea level. After 6 hours pass, the low tide occurs at 6 feet below the average sea level. In this task, you will model this occurrence using a trigonometric function by using x as a measurement of time. Assume the first high tide occurs at x = 0.

1,What are the independent and dependent variables?

2.

Determine these key features of the function that models the tide:

amplitude

period

frequency

midline

vertical shift

phase shift

3.Create a trigonometric function that models the ocean tide for a period of 12 hours.

4.What is the height of the tide after 93 hours?

Nov 4, 2018
#4
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what function would represent this situation  ,Throughout any given month, the maximum and minimum ocean tides follow a periodic pattern. Last year, at a certain location on the California coast, researchers recorded the height of low tide, with respect to sea level, each day during the month of July. The lowest low tide was first measured on July 11, at -1.4 feet. The highest low tide was first measured on July 4, at 1.8 feet. The average low tide for the month of July was measured to be 0.2 feet.

Nov 5, 2018
#6
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I see these last two re CA tides were reposted and Chris is working on the answers.....

ElectricPavlov  Nov 5, 2018