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7 September, 22:54

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 41 and 69 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 48 degrees?

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  1. 7 September, 23:51
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    (a). The sinusoidal function is

    -14*cos ((π/12) * (t-2)) + 55

    (b). The temperature of 48 ° first occurs at 6.00 a. m.

    Step-by-step explanation:

    To solve the question, we note that an example of a sinusoidal function is a cosine function

    a*cos * (2π/k) t+b = Temperature

    For a 24 hour period, the sinusoidal function becomes

    2π/k = 24 or k = π/12 and

    a = (69 - 41) / 2 = 14 also b = 69 - 14 = 55

    Therefore the sinusoidal function becomes

    14*cos ((π/12) * t) + 55 = Temperature at a particular time of day

    checking we have

    at 6 a. m. 14*cos (π/2) + 55 = 55 ° okay

    However the average temperature supposed to occur at 8 a. m.

    Therefore we have time adjustment by 2 hours hence

    Our equation becomes

    -14*cos ((π/12) * (t-2)) + 55 = Temperature

    Therefore at 8 a. m. we have

    -14*cos ((π/12) * (8-2)) + 55 = 55 ° = average temperature

    And

    (b) For 48 °, we have

    -14*cos ((π/12) * (t-2)) + 55 = 48

    or t = 6 a. m.
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