Ferris Wheel Height Model and Ride Timing
A Ferris wheel has a radius of 12 m and its centre is 14 m above the ground. Riders board at the lowest point, which is 2 m above the ground. The wheel rotates once every 40 seconds in a clockwise direction. A rider's height h metres above the ground after t seconds can be modelled by a sinusoidal function of the form h(t) = a sin(b(t - c)) + d. The ride lasts 3 minutes. A safety sensor records a rider's height as 20 m at two different times during the first rotation. The operator wants to know how long during the ride a rider is above 20 m, and whether the model remains valid for the full 3 minutes.
1. Determine the values of a, b, c and d in the model h(t) = a sin(b(t - c)) + d, given that the rider starts at the lowest point at t = 0.
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The centre height gives the vertical shift: d = 14 m. The amplitude is the radius: a = 12 m. Since the rider starts at the minimum (2 m), a negative cosine model is natural, but a sine model requires a phase shift. Using h(t) = a sin(b(t - c)) + d with a = 12 gives a maximum when b(t - c) = π/2. The period is 40 s, so b = 2π/40 = π/20. At t = 0 the height is 2 m, the minimum, so sin(b(0 - c)) = -1, meaning -bc = -π/2, so c = (π/2)/b = (π/2)/(π/20) = 10 s. Thus h(t) = 12 sin(π/20 (t - 10)) + 14. Check: at t = 0, h = 12 sin(-π/2) + 14 = 2 m; at t = 10, h = 12 sin(0) + 14 = 14 m; at t = 20, h = 12 sin(π/2) + 14 = 26 m. The model is consistent with the stated geometry.
2. Find all times in the first 40 seconds when the rider is exactly 20 m above the ground, and hence determine the total time during the 3-minute ride that the rider is above 20 m.
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Set 12 sin(π/20 (t - 10)) + 14 = 20, so sin(π/20 (t - 10)) = 0.5. The general solution is π/20 (t - 10) = π/6 + 2kπ or 5π/6 + 2kπ. For the first case: t - 10 = 20/6 = 10/3, so t = 13.33 s. For the second: t - 10 = 100/6 = 50/3 ≈ 16.67, so t = 26.67 s. Within one 40 s cycle the rider is above 20 m between t = 13.33 s and t = 26.67 s, a duration of 13.33 s. The ride is 180 s, which is 4.5 cycles. In each full cycle the above-20 m interval is 13.33 s, so 4 full cycles give 53.33 s. The remaining 20 s is half a cycle starting at the minimum, so it contains the interval from 13.33 s to 20 s, i.e. 6.67 s. Total = 53.33 + 6.67 = 60 s. The model remains valid because the wheel rotates uniformly and the geometry is fixed; the 3-minute ride is 4.5 rotations, so the sinusoidal model applies throughout.