precalculus

A Ferris wheel is 25 meters in diameter and boarded from a platform that is 2 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 4 minutes. How many minutes of the ride are spent higher than 21 meters above the ground?

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  1. aim for an equation of the type
    height = a sin k(t + d) + c
    where the variables are probably defined in your text or your class notes.

    25 meters in diameter ----> a = 12.5
    minimum is 2 ----> c = 13.5
    1 full revolution in 4 minutes ----> 2π/k = 4, ----> k = π/2

    so lets start with
    h = 12.5 sin π/2(t + d) + 13.5
    when t = 0 , we want h = 2
    12.5 sin π/2(0 + d) + 13.5 = 2
    sin π/2(d) = -.92
    using my calculator:
    π/2(d) = -1.1681..
    d = -.7436..

    height = 12.5sin π/2(t - .7436) + 13.5
    looks good:
    https://www.wolframalpha.com/input/?i=y+%3D+12.5sin(%CF%80%2F2(x+-+.7436))+%2B+13.5

    Now I leave it up to you to solve:
    12.5sin π/2(t - .7436) + 13.5 > 21

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    Reiny
  2. Draw Cartesian system.

    Height from the origin of Cartesian system to center of a circle = height of a platform + radius of the wheel

    h = 2 + 12.5

    h = 14.5 m

    At point x = 0 , y = 14.5 draw a circle whose radius is:

    r = 25 / 2 = 12.5 m

    If total height of the cabin > 21 m then height above horisontal axis of a circle must be:

    h > 21 - 14.5

    h > 6.5 m

    Angular speed:

    ω = angle / tme

    ω = 360° / 4 min

    ω = 90° / min

    t1 = time the wheel cabin takes to reach the starting position from 90°:

    t1 = 90° / ω = 90° / ( 90° / min ) = 1 min

    Mark the angle between the horizontal axis of a circle and the wheel cabin with θ.

    Now:

    sin θ = y / r

    sin θ = y / 12.5

    y = 12.5 ∙ sin θ

    12.5 ∙ sin θ = 6.5

    sin θ = 6.5 / 12.5

    sin θ = 0.52

    t2 = time spent by the cabin to reach the position h = 6.5 m

    ω = angle / tme

    ω = θ / t2

    θ = t2 ∙ ω

    Now you must solve:

    12.5 ∙ sin θ = 6.5

    12.5 ∙ sin ( t2 ∙ ω ) = 6.5

    sin ( t2 ∙ 90° / min ) = 6.5 / 12.5 = 0.52

    ( t2 ∙ 90° / min ) = sin⁻¹ ( 0.52 )

    t2 ∙ 90° / min = 31.3322515°

    t2 = 31.3322515° / ( 90° / min ) = 0.348136127751 min

    Total time for height > 21 m

    t > t1 + t2

    t > 1 + 0.348136127751

    t > 1.348136127751min

    P.S.
    Sorry for my bad English.

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