Ferris Wheel Question

Question

As you ride the Ferris wheel, your distance from the ground varies sinusoidally with time. Let t be the number of seconds that have elapsed since the Ferris wheel started and let d be the number of metres you are from the ground. Assume that you start the ride at the bottom of the Ferris wheel, which is 1.3 metres from the ground. The wheel makes a revolution once every 34 seconds. The diameter of the wheel is 40 metres. What is your height above ground when you have been riding for 7 seconds? Round your answer to the nearest whole number.

Answer 1

Both sine and cosine functions can be used to model this type of situation. But, since you are starting from the bottom and going up, it seems cosine would be best. Since y = cos(x) starts at the top and goes down, if we use a y = -cos(x) as our basic equation, we will get a function that starts at the bottom and goes up.

The generic form of a sinusoidal equation like this would be:

y = -Acos[B(x - C)] + D where A is the amplitude (distance from center to either top or bottom), B is 2π/Period, C is the shift in the "x" direction (in this case that is zero), and D is the distance from the ground (zero) to the center.

So, since the diameter is 40, the radius is 20 meaning A = 20.

B = 2π/34 = π/17

C = 0

D = 20 + 1.3 =21.3

So the function is y = -20cos((π/17)x) + 21.3

Since the independent variable in this function is time, it would be better to use "t" instead of "x" and since this is a function where height is a function of time, let's call it h(t) so:

h(t) = -20cos((π/17)t) + 21.3

To find the height after 7 seconds, plug in 7 for "t":

h(7) = -20cos((π/17)7) + 21.3 = 15.827 ≈ 16 meters high