Ferris Wheel Lighting Budget
A council is designing a Ferris wheel for a harbour festival. The wheel has radius 18 m and its centre sits 20 m above the boarding platform. Riders board at the lowest point of the wheel, which is 2 m above the platform. The wheel rotates once every 90 seconds in a clockwise direction. A lighting engineer wants to switch on a decorative ring of lights only while a marked cabin is more than 30 m above the platform. The council also wants the ride to complete at least 40 full rotations during a 70-minute festival session, and the operator claims this is easily achieved. A safety officer notes that the cabin's height can be modelled as h(t) = 20 - 18 cos(2πt/90), where t is seconds after boarding.
1. Determine the total time per rotation during which the marked cabin is more than 30 m above the platform. Show the inequality and the solution clearly.
Read the worked answer
We need h(t) > 30, so 20 - 18cos(2πt/90) > 30.
Rearrange: -18cos(2πt/90) > 10, hence cos(2πt/90) < -10/18 = -5/9.
Let θ = 2πt/90. Solve cos θ = -5/9. The reference angle is arccos(5/9) ≈ 0.9828 rad. Since cosine is negative in the second and third quadrants, θ ≈ π - 0.9828 ≈ 2.1588 and θ ≈ π + 0.9828 ≈ 4.1244.
So cos θ < -5/9 when 2.1588 < θ < 4.1244. The angular width is 4.1244 - 2.1588 = 1.9656 rad.
Convert to time: t = θ·90/(2π), so duration = 1.9656 × 90/(2π) ≈ 28.2 seconds.
So the lights are on for roughly 28 seconds of each 90-second rotation.
2. Test the operator's claim about 40 rotations in 70 minutes, and comment on whether the claim is valid.
Read the worked answer
Each rotation takes 90 seconds = 1.5 minutes. In 70 minutes the number of rotations is 70/1.5 ≈ 46.7.
Since 46.7 > 40, the operator's claim is valid, with about 6.7 rotations to spare. However, the ride cannot start a new rotation partway through boarding, so realistically the wheel completes 46 full rotations, still exceeding 40.
A judgement call: the claim is true but the margin is modest. If boarding time or a safety pause of more than about 10 minutes total is added, the count could drop toward 40, so the claim is not as comfortable as 'easily achieved' suggests.