Question

At the amusement park, you decide to ride the Ferris wheel, which has a maximum height of 100 meters and a diameter of 50 meters. It takes the wheel seven minutes to make one revolution. Write the sinusoidal function, f(t), that models the height of your chair at any time,

Answer 1

we know a = 50, and the period is 7 minutes
2π/k = 7
k = 2π/7
then let's start with the basic shape of
height = 50sin (2π/7)t , where t is in minutes and height is in metres.
but the maximum height is 100 m , so we have to raise our curve 50 m
height = 50sin (2π/7)t + 50
but we probably want our height to be zero when t = 0, so we need a phase shift
height = 50sin(2π/7)(t+d) + 50
50sin(2π/7)(d) + 50 = 0
sin (2π/7)(d) = -1
we know sin 3π/2 = -1 , so
(2π/7)(d) = 3π/2
d = (3π/2)(7/2π) = 21/4

height = 50 sin (2π/7)(t + 21/4) + 50

check:
if t = 0, height = 0 , check!
if t = 1.75, height = 50
if t = 3.5, height = 100
if t = 5.25 , height = 50
if t = 7 , height = 0, all checks out

f(t) = 50 sin ( (2π/7)(t + 21/4) ) + 50

confirmation:
www.wolframalpha.com/input/?i=plot+y+%3D+50+sin+(+(2%CF%80%2F7)(x+%2B+21%2F4)+)+%2B+50