Solve Ferris Wheel Problem

Question

A Ferris wheel is boarded from a platform that is 5 meters above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The function H ( t ) = − 10 cos ( π /41 t ) + 15 gives your height in meters above the ground t seconds after the wheel begins to turn.How many seconds of the ride (1 full revolution) are spent higher than 18 meters above the ground? The diameter is 20 meters and it takes 82 seconds for one full revolution

Answer 1

So you'd simply need to solve for what angles give a height of 18 and then everything between those two heights is the time for being at least that high.

-10cos(πt/41) + 15 = 18

cos(πt/41) = -0.3

This angle occurs at the angles measuring about 1.8755 and 4.4077 radians (about 107.46 and 252.54 degrees respectively). So simply set πt/41 equal to both of these and solve for t. I found that t = 24.4766 seconds and 57.5236 respectively. So the difference between these, 33.05 seconds, is the time per rotation that the Ferris wheel rider is higher than 18 meters above the ground.

Answer 2

H(t) = - 10cos(π/41t) + 15 > 18; cos(π/41t) < - 0.3; cos^-1(- 0.3) < πt/41 < 2π - cos^-1(πt/41);

1.8755 < πt/41 < 4.4077; 24.48 < t < 57.52