Question

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?

Answer 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..

<b>height = 12.5sin π/2(t - .7436) + 13.5</b>
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

Answer 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.