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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.

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Final answer:

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)

User Ylor
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8.3k points
4 votes

Answer:

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.

User Andrei Kovalev
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