Trig

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A Ferris wheel is 40 meters in diameter and boarded from a platform that is 4 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 2 minutes. How many minutes of the ride are spent higher than 30 meters above the ground?

  • Trig -

    let's use a sine function ....
    amplitude = 20
    period = 2π/k = 2
    2k = 2π
    k = π

    so far we have
    height = 20 sin π(t + c) + d , assuming we have a phase shift and a vertical shift
    clearly, the whole sine curve has to shifted upwards by 24 units, so that the normal min of -20 becomes a min of 4
    so we have
    height = 20 sin π(t+c) + 24

    1. when t = 0 , we want height to be 4
    2. when t = 1 , we want the height to be 44

    1.
    4 = 20sin π(c) + 24
    -20 = 20sin πc
    sin πc = -1
    but we know that sin 3π/2 = -1
    then cπ = 3π/2
    c = 3/2

    equation is
    height = 20 sin π(t + 3/2) + 24

    check for t = 1, (should get 44)
    height = 20 sin π(1 + 3/2) + 24
    = 20 sin 5π/2 + 24 = 20(1) + 24 = 44

    all looks good.

    so we want h ≥ 30
    20 sin π(t+3/2) + 24 = 30
    sin π(t+3/2) = 6/20 = 3/10 = .3
    setting my calculator to radians , I get
    π(t+3/2) = .3047 or π-.3047 = 2.837
    t + 1.5 = .3047/π or t+1.5 = 2.837/π
    t = -1.403 or t = -.597

    BUT, we should have positive times,
    since the period of the curve is 2, adding 2 to each of these times will also satisfy the equation
    so t = -1.403+2 = .597
    and t = -.597 + 2 = 1.403

    so for the time from t = .597 to t = 1.403 the height will be ≥ 30

    So it will be above 30 for( 1.403-.597) minutes
    = .806 minutes or appr 48.4 seconds

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