Precalculus Trigonometry Ferris Wheel Question

Question

A Ferris wheel is 12 meters in diameter and makes one counterclockwise revolution every 6 minutes. Given that riders board the Ferris wheel at ground level, how long does it take for a rider to go from ground level to a height of 9 meters?

It takes____Minutes

Answer 1

This is better explained with a diagram, but I will try to make it as clear as possible.

Draw a circle that represents the Ferris wheel.

Draw a tangent to the circle at the very bottom of the circle. This represents the ground. Label it G.

Place a point on the upper right quadrant of the circle. Label it P.

Draw a perpendicular from P to the ground tangent line. Label it Q.

Draw a line from the center of the circle labelled O to the point P and also from O to G.

Draw a line from P to G which is a chord of the circle.

Finally draw a line from O perpendicular to the chord and label the intersection of this line and the chord M.

1 The line OM bisects the chord and the angle POG.

2 Call the angle PGQ θ.

3 The angle OGQ is 90 - θ.

4 The triangle POG is equilateral; therefor, the angle OPM is also 90 - θ.

5 Angle POG is 180 -2(90 - θ) or 2θ, and the angles POM and MOG are θ.

6 As the chord is bisected, call each bisected segment c.

7 Sinθ = c/6 as 6 is the radius of the wheel in meters.

8 c = 6sinθ, 2c = 12sinθ which is the length of the chord.

9 Sinθ is also equal to PQ/2c as the triangle PGQ is a right angle. PQ is 9 m, the height of the chair at time t.

10 As sinθ = PQ/2c, sinθ = PQ/12 sinθ from step 8.

11 PQ = 12 sin²θ but PQ is 9 m according to the problem.

12 sin²θ = 9/12 = 3/4.

13 sinθ = √3/4 = 0.866

14 θ = 60º

Now to find the time t.

15 The Ferris wheel makes one revolution in 6 minutes or 360/6 = 60 º/min or 60t degrees in t minutes.

16 But we know that angle POG is = 2θ or 120º.

17 From 15 the angle turned from G to P is 60t which is = 120º.

18 So, 60t = 120º and t= 2 min

I recommend that you draw the picture and verify the above steps.

Answer 2

First off you need to derive the equation for the motion of the Ferris wheel. At time zero height is 0 . It will rise to 12 meters and then return to 0 to complete one revolution. Since you start at your minimum value this will be a negative cosine graph. If you can’t see this graph y= -cos(x) and you will see what I am talking about

The midline(vertical shift) and amplitude are both 6 and the period (b) is pi/3 b=2pi/6=pi/3

So equation is y=-6cos(pi/3)t +6

If you want to find how long it takes for the rider to be at 9 meters you need to solve for t when y =9

9 = -6cos(pi/3)t +6. 3=-6cos(pi/3)t. (-1/2) = cos(pi/3)t. Now take inverse cos (arc cos) of both sides

(arc cos -1/2)= (pi/3)t

t= 3(arccos-1/2) / pi

t= 2minutes