help with ythis precalc question

Question

A ferris wheel is 49 meters in diameter and boarded from a platform that is 4 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 8 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn.

What is the amplitude?_____ meters

What is the equation of the Midline?_____   meters  

What is the period?_____   minutes

The equation that models the height of the ferris wheel after  minutes is: _____   

How high are you off of the ground after 7 minutes? Round your answer to the nearest meter._____  meters

Answer 1

The amplitude of a function (in this case going in a circle on the ferris wheel) is found by dividing the distance between the minimum and maximum point by 2. Here, that distance is the diameter, or 49m. The amplitude then would be 24.5m.

The equation of the midline for this ferris wheel is the equation that goes through the center while being parallel to the ground. Since the center of the ferris wheel is halfway up, we add the amplitude to the starting height of 4m to get 24.5m + 4m = 28.5m. That means, the equation for the midline would be h = f(t) = 28.5.

The period of revolution is the amount of time it takes to complete one cycle of the wave. For this example, that's the time it takes to go from your starting point, all the way around and back to where you started. This time is given to you as 8 minutes.

Now that we have the amplitude, midline and period of revolution, we can find the equation that will describe your motion on the ferris wheel.

Because at time 0, you're at the bottom of the ride, so we're at the negative extreme. The trig function that starts at the negative extreme would be -cos(t). We know that we're going to be one amplitude away from the center line at time = 0, so then we must start at -24.5cos(t).

Now that we have the height of our function, let's add the period. Translated into the trig function for cosine where (2*π)/b = period, we solve for b to find b = (2π)/8 = π/4. Let's add that to what we have:

h(t) = -24.5cos([π/4]t)

The last step is to add the midline height so we know where the cosine function is moving around. To add this vertical displacement, add the equation of the midline we found before!

h(t) = -24.5cos([π/4]t) + 28.5

Now to solve for t = 7min we simply plug 7 into the function above:

h(7) = -24.5cos([π/4]*7) +28.5 = 11.17 ≈ 11m

Quick sanity check here: If it takes 8 minutes to do a full circle, at 7 minutes, we should be near the end of the ride. If the end of the ride is 4m above the ground, we should definitely be closer to 4m than to the top of the ride - 53m. Our answer of 11m makes sense.

Hope this helped!

Answer 2

The basic equation for the height of a Ferrris wheel car is:

Acos(Bt)+C

A is the Amplitude of the cosine wave.

B is the angular velocity = 2π/(Minutes per revolution) = 2π/8 = π/4

C is the vertical offset

At t = 0, the A+C = 4.

At t = 4 (half way): Acos(π)+C = -A+C =49+4= 53

Adding the two equations 2C= 57 -->C=57/2 = 28.5 and A=-24.5

The Amplitude for cosine is -24.5

The midline equation = radius plus the platform height = 24.5 + 4 = 28.4 meters

The period is 8 minutes

f(t) = -24.5cos(πt/4) + 28.5

f(7) = -24.5cos(7π/4) + 28.5 = -17.32 + 28.5 = 11.18 ≅ 11 meters