Ferris Wheel and a Polar Sensor
A Ferris wheel has radius 12 m and center 14 m above the ground. A rider boards at the lowest point and the wheel turns counterclockwise at a constant rate, completing one revolution every 40 seconds. A ground sensor tracks the rider's height h(t) in meters. Separately, an engineer describes the rider's position relative to the center using polar coordinates (r, θ) with the center as origin and θ measured from the positive x-axis (east). At t=0 the rider is directly below the center; the wheel rotates counterclockwise. The engineer claims the polar angle θ increases linearly and that r is constant, so the polar description is simpler than the height function.
1. Write h(t) for the rider's height and find all times in the first 40 seconds when the rider is 20 m above the ground.
Read the worked answer
Center height 14, radius 12. Angular speed ω = 2π/40 = π/20 rad/s. At t=0 rider is at bottom, so vertical offset = −12cos(ωt) (starts at −12, rises). Thus h(t) = 14 − 12cos(πt/20).
Set h=20: 14 − 12cos(πt/20) = 20 ⇒ −12cos(πt/20) = 6 ⇒ cos(πt/20) = −1/2. So πt/20 = 2π/3 or 4π/3 (within [0,2π]). Then t = (20/π)(2π/3)=40/3 ≈ 13.3 s, and t = (20/π)(4π/3)=80/3 ≈ 26.7 s. Both are in [0,40]. So the rider is 20 m high at about 13.3 s and 26.7 s.
2. Evaluate the engineer's polar claim and connect θ(t) to the height function.
Read the worked answer
Polar with center as origin: r = 12 constant; θ increases linearly: θ(t) = θ₀ + ωt. At t=0 rider is directly below center, so θ₀ = −π/2 (or 3π/2). Then θ(t) = −π/2 + (π/20)t.
The height above center is r sinθ = 12 sin(−π/2 + πt/20) = −12cos(πt/20), matching the offset in h(t). So the polar description is simpler: r constant, θ linear, and the height is just 14 + r sinθ. The engineer is correct that the polar form isolates the constant radius and uniform angular motion; the height function is a cosine projection of that circular motion. The claim holds because the wheel rotates at constant angular speed, making θ linear in t.