Exemplo 13 de Algebra and Trigonometry, 8.1 Graphs of the Sine and Cosine Functions

Determining a Rider’s Height on a Ferris Wheel

Problema e solução em inglês, como no livro original.

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.

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

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Solução
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 $B = \frac{2 π}{30} = \frac{π}{15}$
- Shape: $−cos (t)$
An equation for the rider’s height would be

$y = - 67.5 \cos (\frac{π}{15} t) + 69.5$

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

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Solução