Modeling with Trigonometric Functions

Question

A Ferris wheel is 40 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 34 meters above the ground?

Answer 1

The y coordinate of a point on a circle is related to its center and the angle by y_final = y_center + R*sin(θ) where R is the radius. For this problem we know that R = 20 and y_center = R + 2 = 22.

Now all we need to do is calculate the θ value that y_final = 22 + 20*sin(θ) is greater than 34.

sin^-1(12/20) = θ = 36.87 degrees. Because of the identity: sin(θ) = sin(180-θ), θ also can equal 143.13 degrees.

To find the amount of degrees where the rider is above 34m we take 143.13 - 36.87 = 106.26 degrees.

Next, we take 106.26/360 to find the percent of the time that the rider is above 34m to get 29.5%.

Finally we take 10 min * 29.5% to get our final answer of 2.95 minutes.