Questions:

As you are waiting in line to ride the Ferris wheel at the state fair, you begin to be hypnotized by the repetitive motion of the seats. The seat is at the bottom; it's halfway up, it's at the top; it's halfway down; it's at the bottom; it's halfway up; it's at the top...

You decide to time the wheel and observe that the ferris wheel makes 4 complete revolutions every minute. You also assume that the diameter of the wheel is 100 feet because a sign beside the wheels says "Climb 100 feet into the sky on our Ferris wheel". However, you note that when you get into a seat, the seat is about 3 feet off the ground, so technically the maximum height of a person would reach would be...?

a) Write two equations, one sine and one cosine, that would represent the distance above the ground as a function of time. Let t = 0 when the seat is at the lowest point on the wheel.

b) How high will a person be 10 seconds after they begin riding the ferris wheel?

c) How long after a person gets on the ride will they actually be 100 feet in the air?

d) How long, during each revolution, is a person more than 60 feet in the air?

Answers:

a)

Sine

A) 50

B) (2π) / 15

C) 3.75

D) 53

Cosine

A) -50

B) (2π) / 15

C) 0

D) 53

Sine : d (t) = 50 sin [ ((2π) / 15) (t - 3.75)] + 53

Cosine : d (t) = -50 cos [ ((2π) / 15) (t)] + 53

b)

d (t) = -50 cos [ ((2π) / 15) (t)] + 53

d (t) = -50 cos [ ((2π) / 15) (10)] + 53

d = 78ft

c)

d (t) = -50 cos [ ((2π) / 15) (t)] + 53

100 = -50 cos [ ((2π) / 15) (t)] + 53

let θ = [ ((2π) / 15) (t)]

100 = -50 cos θ + 53

47 = -50 cos θ

cos θ = (-0.94)

θ = 2. 7934 , 3.4898

((2π) / 15) (t) = 2.7934

t = 2.7934 (15 / (2π))

t = 6.6688 s

After a person gets on the ride they will be 100 feet in the air for 6.67 s.

d)

d (t) = -50 cos [ ((2π) / 15) (t)] + 53

60 = -50 cos [ ((2π) / 15) (t)] + 53

let θ = [ ((2π) / 15) (t)]

60 = -50 cos θ + 53

7 = -50 cos θ

cos θ = (-0.14)

θ = 1.71126 , 4.5719

((2π) / 15) (t) = 1.71126

t = 1.71126 (15 / (2π))

t = 4.0853 s

((2π) / 15) (t) = 4.5719

t = 4.5719 (15 / (2π))

t = 10.9146

10.9146 - 4.0853 s = 6.8293 s

In each revolution a person is above the ground for 60 feet for 6.83 s.

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