A carnival Ferris wheel with a radius of 9m rotates once every 10 s. The bottom of the wheel is 1m above the ground.

Question

a) Sketch the graph showing two complete cycles if the rider starts at the bottom of the wheel.

b) Find the equation of the sine function that gives a rider’s height above the ground, in metres, as a function of the time, in seconds, with the rider starting at the bottom of the wheel.

Answer 1

9 = amplitude = A

10 = period

h=9+1 = 10 is the midline = D

period= 2pi/B = 10

B=2pi/10 = pi/5

C is the phase shift that makes h(0) = 1, C=-pi/2

h(t) = Asin(2pi/B)(t) + C) + D

h(t) = 9sin(2pi/5)t -pi/2) + 10

or

h(t) = 9cos(2pi/5)t + pi) +10

in 2 1/2 seconds or 7 1/2 seconds the height = 10

in 5 seconds the height = 19

in 10 seconds the height = 1

in 12.5 and 17.5 seconds the height is 10

in 15 seconds the height = 19

in 20 seconds the height = 1

for a 2nd cycle

h(t) oscillates from 1 to 10 to 29 to 10 to 1 to 10 to 29 to 10 to 1 etc. with a smooth curve connecting those points (0,1), (2.5,10), (5,29), (7.5,10), (10,1), (12.5, 10), (15, 29), (17.5, 10), (20, 1). h on the vertical axis, t on the horizontal axis