parametric equations?

Question

A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the the x-axis. The curve traced out by P is given by the following parametric equations:
x = 8*theta - 6*sin(\theta)
y = 8 - 6*cos(theta)

What must we have for R and d?

Answer 1

The parametric equation for a trochoid are:
x = a*theta - b*sin(\theta)
y = a - b*cos(theta)

Where a = radius of circle = R
b = distance of point from centre = d

Hence
x = R*theta - d*sin(\theta)
y = R - d*cos(theta)

Comparing with given equation:

R=8
d=6

Hope this helps!

Answer 2

You can visualize it by moving the platform to the left and keeping the wheel center still. In this case the point P will have x = - d*sin(theta), y = R - d*cos(theta), where theta is the angle of wheel rotation. Now if you keep platform still the wheel moves along the x axis for R*theta and the point P with it. It changes x = R*theta - d*sin(theta). If you compare these with your formulas yo get R = 8, d = 6.