A Ferris wheel 80 meters in diameter turns continuously, completing a single rotation once every 8 minutes. You must board the Ferris wheel at its lowest point from a platform located 5 meters above the ground. If you board the Ferris wheel at t=0, write and sketch a sinusoidal function to model your height above the ground after t minutes on the Ferris wheel. How high above the ground will you be after 5 minutes? How many minutes into the ride will it be when you first reach 35 meters above the ground?

Step-by-step explanation:

General form of a sine or cosine wave is:

y = ±A sin((2π/T) t) + B

y = ±A cos((2π/T) t) + B

where A is the amplitude, T is the period, and B is the vertical offset or midline.

The Ferris wheel is 80 meters in diameter or 40 meters in radius.

A = 40

The Ferris wheel turns once every 8 minutes.

T = 8

The lowest point is 5 meters above the ground, so the center of the Ferris wheel is 45 meters above the ground.

B = 45

At t=0, you're at the lowest point, so use -cos.

Therefore, the equation is:

y = -40 cos(2π/8 t) + 45

y = -40 cos(π/4 t) + 45

When t = 5:

y = -40 cos(5π/4) + 45

y = 20√2 + 45

y ≈ 73.3 meters

When y = 35:

35 = -40 cos(π/4 t) + 45

-10 = -40 cos(π/4 t)

1/4 = cos(π/4 t)

π/4 t ≈ 1.318

t ≈ 1.68 minutes

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