Question

a ferris wheel has a radius of 10m and is one meter above the ground. If the ferris wheel
makes 1 revolution every 20 seconds, write an equation that gives the height above the
ground of a person on the ferris wheel as a function of time if that person starts (t=0)
1/8th of a ..

Answer 1

I think your sentence was cut off — do you mean "1/8 of a revolution from the lowest point"? I'll solve that case; if you meant something else (from the top, or from the rightmost point) tell me and I’ll adjust.

Assume 1/8 revolution above the lowest point, counterclockwise. The center of the wheel is 10 + 1 = 11 m above the ground, the amplitude is 10 m, and the angular speed is ω = 2π/20 = π/10 rad/s. If θ(t) is the standard polar angle and we take the height as
h(t) = 11 + 10 sin(θ(t)),
then the initial angle is θ(0) = −π/2 + 2π(1/8) = −π/4, so θ(t) = (π/10) t − π/4. Thus

h(t) = 11 + 10 sin( (π/10) t − π/4 ) (height in meters, t in seconds).

Quick checks: h(0) = 11 + 10 sin(−π/4) = 11 − 10·(√2/2) ≈ 3.93 m; the period is 20 s (h(t+20)=h(t)).