Tyler took a ride on the Speed Demon, a Ferris wheel at a local carnival. The wheel has a 64-ft diameter and?

minimum height = - 8

amplitude = 32 (half of the diameter)

center of oscillation = -8 + 32 = 24

maximum height = -8 + 64 = 56

period = 60 / 2.5 = 24 seconds

w = 2pi / period = 2pi / 24 = pi / 12

as a sin function, the minimum occurs at 3/4 of a cycle, or 18 seconds into the period

thus, the beginning of the cycle will occur 1/4 of a cycle later, or 6 seconds after the minimum when the function is back to the center of oscillation going up

this will give us a phase shift of 6 seconds to the right (equivalent to 18 seconds to the left)

h(t) = 32 sin [pi/12 (t - 6)] + 24 (where t is in seconds)

check: h(0) = 32 sin (-pi/2) + 24 = -8, a min

h(6) = 32 sin 0 + 24 = 24, the center of oscillation

h(12) = 32 sin (pi/2) + 24 = 56, the max value

h(18) = 32 sin (pi) + 24 = 24, the center of oscillation again

h(24) = 32 sin (3pi/2) + 24 = -8, back to the min

h(t) = 0 ==> 32 sin [pi/12(t - 6)] + 24 = 0

sin [pi/12 (t-6)] = -24/32 = -3/4

pi/12 (t - 6) = sin^-1 (-3/4)

pi/12 (t - 6) = -.848

t - 6 = -.848 (12/pi) = -3.24

t = 6 - 3.24 = 2.76

Tyler will first reach ground level after about 2.76 seconds

note: if you had a choice, you could have written this as a negative cos curve with no phase shift:

h(t) = -32cos (t pi/12) + 24

this automatically would give you a min a t = 0 (where the negative cos curve starts)