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Math Help - Sinusoidal Graph question

  1. #1
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    Sinusoidal Graph question

    Hey everyone, I'm at a question that I'm a little stumped on the method of solving, and there is no solution guide in the textbook .

    4. In a New Zealand city the time of sunrise can be found using the trigonometric function

    T= 4.5sin(pi(d-4)/180)+4.9

    where T is the number of hours past midnight d and is the number of days into the year. Find the time to the nearest minute when the
    Sun will rise on the 7th day of the year. How many days in the year does the sunrise after 7 am?

    The bolded part of the question is the part I am not sure how to solve.

    Any help?
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Sinusoidal Graph question

    Days of the year when sunrise is after 7 a.m. are:

    {32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156}

    which are 125 days.



    This can't be NZ
    Attached Thumbnails Attached Thumbnails Sinusoidal Graph question-nz-sunrise.png  
    Last edited by MaxJasper; September 27th 2012 at 09:08 PM.
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: Sinusoidal Graph question

    You want to solve:

    4.5\sin\left(\frac{\pi(d-4)}{180} \right)+4.9>7

    4.5\sin\left(\frac{\pi(d-4)}{180} \right)>2.1

    \sin\left(\frac{\pi(d-4)}{180} \right)>\frac{7}{15}

    So we want:

    \sin^{-1}\left(\frac{7}{15} \right)<\frac{\pi(d-4)}{180}<\pi-\sin^{-1}\left(\frac{7}{15} \right)

    \frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)<d-4<180-\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)

    \frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)+4<d<184-\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)

    You should find, using your calculator that (approximately):

    31.8<d<156.2

    Since d is an integer, we may write:

    32\le d\le156

    Hence, the number of days N is:

    N=(156-32)+1=125
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