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math-precalculus

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 73 and 87 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 77 degrees?

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  1. the amplitude is (87-73)/2 = 7
    the center line is (87+73)/2 = 80

    y = 7sin(k(x-h))+80

    since y(8) = 0,

    y = 7sin(k(x-8))+80

    since the period is 24 hours, 2π/k=24, so k = π/12

    y = 7sin(π/12 (x-8))+80

    so, when is y=77?

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  2. Substitute y for 77 Get sin by itself, divide by 7 on both sides

    Consider everything in the parenthesis of sin as theta, then solve using sin^-1

    77 = 7sin(theta)+80
    -3 = 7sin(theta)
    -3/7 = sin(theta)
    or
    sin^-1(-3/7)=theta

    take the answer to sin^-1(3/7) and then operate on it with what theta was , (π/12 (x-8))
    get x by itself, multiply by the reciprocal 12/π to both sides and then add 8.

    That should be your answer. Hope that helped.

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