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

To model Amare's height above the ground as a function of time, use the equation h(t) = 2 + 2 * sin((2 * pi * t) / 6)

Step-by-step explanation:

To determine a function to model Amare's height above the ground as a function of time, we can use the equation of motion for circular motion. The height of Amare above the ground can be calculated using the equation:

h(t) = r + R * sin(theta)

Where:

h(t) is the height above the ground at time t

r is the radius of the wheel (half the diameter)

R is the maximum height above the ground (given as 4 meters)

theta is the angle in radians, which is related to time by the formula theta = (2 * pi * t) / T, where T is the time for one revolution (given as 6 minutes)

Putting all these values into the equation, we get:

h(t) = 2 + 2 * sin((2 * pi * t) / 6)

Answer 2

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

Step-by-step explanation:

The general form of cosine function is

`y=A\cos(Bt+C)+D`

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

It is given that the 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 is 50+4=54.

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

It takes six minutes to do three revolutions on the Ferris wheel. So, period of the function is

`period=(6)/(3)`

`period=2`

The period of a cosine function is

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

The function have no phase shift. So, C=0.

Substitute y=h(t), B=π, C=0 and D=29 in equation (1) to find the function.

`h(t)=A\cos(\pi t+0)+29`

It is given that the ride is at the low point whet t=0, it means the function passes through the point (0,4).

Substitute t=0 and h(t)=4 in the above function.

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

`4=A+29`

Subtract 29 from both the sides.

`4-29=A+29-29`

`-25=A`

The amplitude of the function is -25.

Substitute y=h(t), A=-25 B=π, C=0 and D=29 in equation (1) to find the function.

`h(t)=-25\cos(\pi t)+29`

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