John has been hired to design an exciting carnival ride. Tiff, the carnival owne

Question

John has been hired to design an exciting carnival ride. Tiff, the carnival owner, has decided to create the world's greatest Ferris wheel. Tiff isn't into math; she simply has a vision and has told John these constraints on her dream: (i) the wheel should rotate counterclockwise with an angular speed of a = 13 RPM; (ii) the linear speed of a rider should be 200 mph; (iii) the lowest point on the ride should be c = 4 feet above the level ground. The wheel starts turning when Tiff is at the location P, which makes an angle with the horizontal, as pictured. It takes her 1.5 seconds to reach the top of the ride. (Impose coordinates with units in feet.)

(a) Impose a coordinate system with the origin at the center of the wheel. Find the coordinates

T(t) = (x(t), y(t))

of Tiff at time t seconds after she starts the ride. (Round values to three decimal places as needed.)

(b) Tiff becomes a human missile after 6 seconds on the ride. Find Tiff's coordinates the instant she becomes a human missile. (Round your answers to three decimal places.)
(x(6), y(6)) =

Sketch a picture indicating this line and her initial direction of motion along it when the seat detaches.

(c) Find the equation of the tangential line along which Tiff travels the instant she becomes a human missile. (Round values to three decimal places as needed.)
y =  

Explanation / Answer

  Using dimensional analysis to convert from revolutions per minute to miles per hour and solve for the radius:

13 rpm ·( 2?r ft/ 1 revolution)· (1 mile / 5280 ft) · (60 minutes / 1 hour) = 250 miles / hour

(13?r)/44 miles / hour = 250 miles/hour

(13?r)/44 = 250

13?r = 11000

r = 11000 / (13?r)

r = 269.34 ft. ? 538.68 ft diameter

Even though you didn't link to a diagram referred by the question, I will try to answer it anyway.

From P to the top of the wheel takes 1.5 seconds. The period is 60/13 seconds (1 revolution). 1.5 sec / (60/13) = .325 of a revolution. If you subtract ¼ of a revolution from that, you have the arc length intersected by

? .325 - .25 = .075revolution = .075·360° = 27°

Since from P to the top of the wheel takes 1.5 seconds, then from the top of the wheel until Tiff becomes a human missile is 4.6 seconds. 4.6 seconds / (60/13) = .9967 revolutions. 0.9967·360° = 358.80°

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