Question

Amare wants to ride a Ferris wheel that sits four meters above the ground and has a diameter of 50 meters. It takes six minutes to do three revolutions on the Ferris wheel. Complete the function, h(t), which models Amare's height above the ground, in meters, as a function of time, t, in minutes. Assume he enters the ride at the low point when t = 0.

Answer 1

The required function is
`y=-25\cos (\pi x)+29`.

Explanation:

The general form of a cosine function is

`y=A\cos (Bx+C)+D` .... (1)

where, A is amplitude,
`(2\pi)/(B)` is period, C is phase shift and D is mid line.

It is given that Ferris wheel that sits four meters above the ground and has a diameter of 50 meters. So the minimum value of the function is 4 and maximum value of the function 54 meters.

`D=Midline=(Maximum+Minimum)/(2)=(4+54)/(2)=29`

It takes six minutes to do three revolutions on the Ferris wheel.

`Period=(6)/(3)`

`Period=2`

`(2\pi)/(B)=2`

`B=\pi`

Phase shift is not given so C=0.

Substitute B=π, C=0 and D=29 in equation (1).

`y=A\cos (\pi x+0)+29`

`y=A\cos (\pi x)+29` ... (2)

He enters the ride at the low point when t = 0. It means the function passes through the point (0,4).

`4=A\cos (\pi (0))+29`

`4=A+29`

`4-29=A`

`-25=A`

The amplitude is -25. Put this value in equation (2).

`y=-25\cos (\pi x)+29`

Therefore the required function is
`y=-25\cos (\pi x)+29`.

Answer 2

The distance from the bottom of the wheel to a point on the ground can be modeled by a sine function. h(t)=a*sin(kt)+b. b is the height of the wheel, or 4m. a is the radius of the wheel, 50/2=25m. 2pi/k is the 5sinof thw revolution, so 2pi/k=6/3=2, k=pi. h(t)=25sin(pi*t)+4.