Questions: Before being replaced in 2016, a large Ferris wheel on Chicago, Illinois' Navy Pier had a diameter of 140 feet and the riders boarded from a platform 10 feet above the ground. The Ferris wheel completed one rotation every 7 minutes. Express a rider's height, H, in feet above the ground as a function of time, t, in minutes. a.) H(t)=-70 sin (2 pi/7 t)+80 b.) H(t)=-70 cos (2 pi/7 t)+80 c.) H(t)=70 cos (2 pi/7 t)+80 d.) H(t)=70 sin (2 pi/7 t)+10

Before being replaced in 2016, a large Ferris wheel on Chicago, Illinois' Navy Pier had a diameter of 140 feet and the riders boarded from a platform 10 feet above the ground. The Ferris wheel completed one rotation every 7 minutes.

Express a rider's height, H, in feet above the ground as a function of time, t, in minutes.
a.) H(t)=-70 sin (2 pi/7 t)+80
b.) H(t)=-70 cos (2 pi/7 t)+80
c.) H(t)=70 cos (2 pi/7 t)+80
d.) H(t)=70 sin (2 pi/7 t)+10

Transcript text: Before being replaced in 2016, a large Ferris wheel on Chicago, Illinois' Navy Pier had a diameter of 140 feet and the riders boarded from a platform 10 feet above the ground. The Ferris wheel completed one rotation every 7 minutes. Express a rider's height, $H$, in feet above the ground as a function of time, $t$, in minutes. a.) $H(t)=-70 \sin \left(\frac{2 \pi}{7} t\right)+80$ b.) $H(t)=-70 \cos \left(\frac{2 \pi}{7} t\right)+80$ c.) $H(t)=70 \cos \left(\frac{2 \pi}{7} t\right)+80$ d.) $H(t)=70 \sin \left(\frac{2 \pi}{7} t\right)+10$

Solution

Solution Steps

Step 1: Identify the amplitude and vertical shift

The Ferris wheel has a diameter of 140 feet, so the radius (amplitude) is $ \frac{140}{2} = 70 $ feet. The riders board from a platform 10 feet above the ground, so the vertical shift is $ 70 + 10 = 80 $ feet.

Step 2: Determine the period and angular frequency

The Ferris wheel completes one rotation every 7 minutes, so the period $ T = 7 $ minutes. The angular frequency $ \omega $ is given by: \[ \omega = \frac{2\pi}{T} = \frac{2\pi}{7}. \]

Step 3: Choose the correct trigonometric function

The height of the rider starts at the lowest point (10 feet) when $ t = 0 $. This corresponds to a cosine function with a negative amplitude: \[ H(t) = -70 \cos\left(\frac{2\pi}{7} t\right) + 80. \]

Step 4: Verify the correct option

The correct function matches option b: \[ H(t) = -70 \cos\left(\frac{2\pi}{7} t\right) + 80. \]