PreCalc

The temperature T(t) varies sinusoidally on a certain day in December. The minimum temperature is 35 degrees Fahrenheit at midnight. The maximum temperature is 50 degrees Fahrenheit at noon. Let t be the number of hours since midnight (t=o at midnight).
a.) Sketch and label a graph showing exactly two periods of T(t)beginning at t=0.
b.)Determine a function for T(t) using the cosine function.
c.)Determine a function for T(t) using the sine function.
d.) Use your equation(s) to find the temperature at 1 am.
e.) I want to go on a bike ride, but I prefer to ride when the temperature is at least 45 degrees Fahrenheit. What is the earliest time of day that I can leave for my ride? How long can I stay out before I get cold?

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asked by Katy
  1. the max-min value is 50-35=15, so the amplitude is 7.5. The center line is (35+50)/2 = 37.5

    y = 7.5 sin(x) + 37.5

    y has a max at t=12, and the period is 24 hours, so the minimum k is a t t=0.

    cos(x) has a max at x=0, so we have -cos(x) and thus

    y = -7.5 cos(π/12 x) + 37.5

    Now use that to answer the other questions. recall that cos(x) = sin(π/2 - x).

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    posted by Steve
  2. when t=0 , temp = 35
    when t = 12, temp = 50
    Period = 24 hours or k = π/12

    how about
    Temp = 7.5cos (π/12)(t + 12) + 42.5

    check:
    when t = 0, Temp = 7.5cos(π) + 42.5 = 35
    when t = 12, Temp = 7.5cos(2π) + 42.5 = 50

    You try it with a sine curve, be aware that your phase shift will have to be different.

    a 1:00 am , t = 1
    temp = 7.5cos (π/12)(13) + 42.5 = 35.26° F

    for T ≥ 45
    7.5cos (π/12)(t+12) + 42.5 = 45
    7.5cos (π/12)(t+12) = 2.5
    cos (π/12)(t+12) = .3333...
    (π/12)(t+12) = 1.231 or (π/12)(t+12) = 2π - 1.231 = 5.052
    t+12 = 4.702 or t+12 = 19.297
    t = -7.298 or t = 7.297 because of the symmetry of the curve
    and
    t = 19.297
    t = 7.297 hrs = appr 7:18 am
    t = 19.297 = appr 7:18 pm

    so the temp is above 45° F from 7:18 am to 7:18 pm

    check my arithmetic

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    posted by Reiny
  3. Reiny is right: 42.5, not 37.5

    but I'm sure you caught my error.

    Note how he offset his function by using a phase shift, which has the same effect as changing the sign as I did, since shifting by 1/2 period does that flip:

    cos(x+pi) = -cos(x)

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    posted by Steve

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