Precalc question

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 12 RPM; (ii) the linear speed of a rider should be 200 mph; (iii) the lowest point on the ride should be 4 feet above the level ground.

(a) Impose a coordinate system and find the coordinates T(t) = (x(t),y(t)) of Tiff at time t seconds after she starts the ride.

(b) Tiff becomes a human missile after 6 seconds on the ride. Find Tiff’s coordinates the instant she becomes a human missile.

(c) Find the equation of the tangential line along which Tiff travels the instant she becomes a human missile.

Answer 1

First convert angular velocity ω in RPM to radians/hour and radians/sec: ω = 12 (rev/min)*(2π rad/rev)*(60 min/hour) = 4,523.9 rad/hour *(1 hour/3600 sec) = 1.2566 rad/sec. Now linear speed v = radius * angular velocity, v = 200 mph = rω = 4,523.9 * r, so radius of Ferris wheel r = (200 mi/hr)*(5,280 ft/mi) / (4,523.9 rad/hr) = 233.4 ft.

a) Let the center of the circle be (0, 237.4) so that the rider clears the ground by 4 ft. Starting at the lowest point, we can let x = r sin(ωt) and y = - r cos(ωt) + 237.4. This yields T(t) = (233.4 sin(1.2566t), -233.4 cos(1.2566t) + 237.4).

b) At 6 seconds, x(6) = 221.96 and y(6) = 165.23 [NOTE: calculator must be in RAD mode, not DEG!]  Tiff has just completed 1 rev and is now in 4th quadrant.

c) Use the above x,y coordinates, relative to the center, to find the slope of the radius vector: m = Δy/Δx = (165.23 - 237.4) / (221.96 - 0) = -0.32515. The tangent line's slope is therefore the negative reciprocal of this slope, so tangent line's slope = +3.0755. With y = mx + b, 165.23 = 3.0755 * 221.96 + b, b = 517.4, and the tangent line's equation is y = 3.0755 x + 517.4.