Student Question

The height of a rider on a ferris wheel, in metres can be modeled using the function H(0)=10sin(0-90•)+12 A) state the amplitude, phase shift and vertical translation B) sketch the function C) what is the radius of te ferris wheel D) what is the height of the rider when the ride begins? E) how would the graph change if the ferris wheel turned in the opposite direction.

Expert Answers

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Given `h(theta)=10sin(theta-90^circ)+12` where h gives the height in meters:

(a) The amplitude is 10. The phase shift is a horizontal translation of 90 to the right. The vertical translation is 12 up.

(if f(x)=AsinB(x-h)+k ; |A| gives the amplitude, if A<0 then reflect over the x-axis; B gives the horizontal stretch/compression [change in period]; h is the horizontal translation; k is the vertical translation)

(b) The sketch:

(The peak in the first quadrant is at 180; the first trough after 0 is at 360)

(c) The radius is 1/2 the diameter of the ferris wheel. The height of the rider ranges from 2 meters to 22 meters, so the diameter of the wheel is 20 meters. The radius is 10m.

(d) When the ride begins at `theta=0` the rider is 2m above the ground.

Look at the graph or plug in `theta=0` . The rider's maximum height is 22m.

(e) The graph will not change if the wheel spins the other direction. Consider the situation: the rider begins at the lowest point, the height increases until the maximum is reached, and the decreases until the minimum is reached, and repeat; regardless of which direction the wheel turns.

Mathematically you could also write the function given as `h(theta)=-10cos(theta)+12` . Since `cos(-theta)=cos(theta)` the function is the same. (`cos(-theta)` describes what happens if the wheel reverses direction)

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