Double Ferris wheel problem
I got this project that I am doing for pre-Calculus and I got question #1 and question 2 but could someone point me in the right directions so i can solve the other problems.
Modeling a double Ferris Wheel
In 1939, John Courtney invented the double Ferris wheel, called a Sky Wheel, consisting of two small wheels spinning at the ends of a rotating arm.
For this project we will model a double Ferris wheel with a 50 foot arm that is spinning at a rate of 3 revolutions per minute in a counterclockwise direction. The center of the arm is 44 feet above the ground. The diameter of each wheel is 32 feet, and the wheels turn at a rate of 5 revolutions per minute in a clockwise direction. A diagram of the situation is shown in figure 1. M is the midpoint of the arm, and O is the center of the lower wheel. Assume the rider is initially at point P on the wheel.
1. Find the lengths MO, OP, and MG.
Found to be
Figure 2 shows the location of the rider after a short amount of time has passed. The arm has rotated counterclockwise through an angle θ, while the wheel has rotated clockwise through an angle φ relative to the direction of the arm. Point P shows the current position of the rider, and the height of the rider is h.
2. Find angle QOP in terms of θ and φ.
Found to be
3. Use right triangle trigonometry to find the lengths a and b, and then the height of the rider h, in terms of θ and φ.
We know that the angular velocity of the arm is 3 revolutions per minute, and the angular velocity of the wheel is 5 revolutions per minute. The last step is to find the radian measure of angles θ and φ.
4. Let t be the number of minutes that have passed since the ride began. Use the angular velocities of the arm and wheel to find θ and φ in your answer from Step 3 to obtain the height as a function of time t (in minutes).
5. Use the function to find the following:
(a) The height of the rider after 30 seconds have passed.
(b) The height of the rider after the arm has completed its first revolution.
(c) The height of the rider after the wheel has completed two revolutions.
6. Graph this function with the calculator. Make sure the calculator is set to radian mode. Use the graph to find the following:
(a) The maximum height of the rider.
(b) The minimal number of minutes required for the rider to return to the original position (point P in Figure 1).