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Math Help - Quick Sinusoidal Problem Help

  1. #1
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    Quick Sinusoidal Problem Help

    Hey guys can you help me with this one question please and maybe explain the process.

    The daylight time (time between sunrise and sunset) varies sinusoidally throughout a year. The longest day of the year is the 170th day with a daylight time of 15 hours. The Shortest daylight time occurs on the 353rd day of the year with a daylight time of 6 hours.

    It's a two part question:

    What is the period and amplitude: I figured A=4.5 and P=365

    And then it asks to create a graph of this model and generate a specific equation with time t=0 being the first day of the year.

    So could you guys explain how to graph this equation? She also said there is a vertical and horizontal shift and that C/B=170.

    I was sick the day my teacher was explaining problems like these and am totally lost.
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  2. #2
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    Where T is total length of the day and D is day of the year you have

    T= A\sin b\left(D-c \right) +d

    You need to find a,b,c,d

    From your information

    T= 4.5\sin \frac{2\pi}{365}\left(D-c \right) +10.5

    Now to find c use the fact that

    Quote Originally Posted by falconskid007 View Post
    The longest day of the year is the 170th day with a daylight time of 15 hours. The Shortest daylight time occurs on the 353rd day of the year with a daylight time of 6 hours.

    What do you get?
    Last edited by pickslides; May 22nd 2010 at 02:39 PM.
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  3. #3
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    skeeter's Avatar
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    y = 4.5 \sin\left[\frac{2\pi}{365}(t - c)\right] + 10.5

    y_{max} = 15 occurs at t = 170, and when \sin\left[\frac{2\pi}{365}(170 - c)\right] = 1

    \frac{2\pi}{365}(170 - c) = \frac{\pi}{2}

    170 - c = \frac{365}{4}

    c = 170 - \frac{365}{4} = \frac{315}{4}

    y = 4.5 \sin\left[\frac{2\pi}{365}\left(t - \frac{315}{4}\right)\right] + 10.5

    graph attached
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