Ferris wheel

Question

The movement of a chair on a Ferris Wheel can be modeled using the equation y = -10 cos (3x) + 11.5 (where x is in seconds and y is in metres). Describe how how the equation relates to the Ferris Wheel in terms of the following:

• how long it takes for one rotation
• the size of the Ferris Wheel’s diameter
• the location of the axle of the Ferris Wheel
• the height of a rider after 45 seconds

Answer 1

y=-10cos(3x)+11.5 is a periodic function. At x=0, y=11.5-10=1.5m and when 3x=360 degrees or 2π radians. So x=120secs or 2π/3secs. The latter would imply a complete rotation in about 2 seconds, which would make for an uncomfortable ride! So we assume a revolution occurs every 120 seconds = 2 minutes, a more leisurely ride.

The highest point reached is when cos(3x)=-1, y=21.5m, i.e., 3x=180, x=60secs=1 minute. The distance from the lowest point (1.5m) to the highest (21.5m) is 20m, the diameter of the wheel. The axle is one radius (10m) plus the minimum height (1.5m) = 11.5m above ground. After 45secs, 3*45=135º, and cos135=-√2/2. So y=11.5+5√2=18.57m approx.