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Determining a rider’s height on a ferris wheel

The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.

With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.

Passengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. The midline of the oscillation will be at 69.5 m.

The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.

Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.

  • Amplitude: 67 .5, so A = 67.5
  • Midline: 69 .5, so D = 69.5
  • Period: 30, so B = 2 π 30 = π 15
  • Shape: −cos ( t )

An equation for the rider’s height would be

y = 67.5 cos ( π 15 t ) + 69.5

where t is in minutes and y is measured in meters.

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Access these online resources for additional instruction and practice with graphs of sine and cosine functions.

Key equations

Sinusoidal functions f ( x ) = A sin ( B x C ) + D f ( x ) = A cos ( B x C ) + D

Key concepts

  • Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of 2 π .
  • The function sin x is odd, so its graph is symmetric about the origin. The function cos x is even, so its graph is symmetric about the y -axis.
  • The graph of a sinusoidal function has the same general shape as a sine or cosine function.
  • In the general formula for a sinusoidal function, the period is P = 2 π | B | . See [link] .
  • In the general formula for a sinusoidal function, | A | represents amplitude. If | A | > 1 , the function is stretched, whereas if | A | < 1 , the function is compressed. See [link] .
  • The value C B in the general formula for a sinusoidal function indicates the phase shift. See [link] .
  • The value D in the general formula for a sinusoidal function indicates the vertical shift from the midline. See [link] .
  • Combinations of variations of sinusoidal functions can be detected from an equation. See [link] .
  • The equation for a sinusoidal function can be determined from a graph. See [link] and [link] .
  • A function can be graphed by identifying its amplitude and period. See [link] and [link] .
  • A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See [link] .
  • Sinusoidal functions can be used to solve real-world problems. See [link] , [link] , and [link] .

Section exercises

Verbal

Why are the sine and cosine functions called periodic functions?

The sine and cosine functions have the property that f ( x + P ) = f ( x ) for a certain P . This means that the function values repeat for every P units on the x -axis.

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How does the graph of y = sin x compare with the graph of y = cos x ? Explain how you could horizontally translate the graph of y = sin x to obtain y = cos x .

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Questions & Answers

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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