Posted by **Anon** on Friday, January 20, 2012 at 3:42pm.

The figure shows part of a curve traced by a point on the circumference of a circle of radius 4 that rotates, without slipping, around a fixed circle of radius 2. The rotating circle starts with angle t = 0 radians and the point P (x, y) at (10, 0). In this problem you will find parametric equations of the resulting epicycloid.

In the figure, t = 0.9 radian. Find the coordinates (rounded to the nearest hundredth) of the center of the large moving circle.

Because the big circle rotates wtihout slipping, arc a on the big circle equals arc a on the small circle. Find a when t = 0.9 radian, as in the figure. Use the answer to find the measure of angle A that subtends arc a on the big circle.

a = units

angle A = radians

Angle θ at the center of the big circle has measure equal to t + A. Find θ when t = 0.9 radian. θ = radians.

Use the answers above to find the coordinates of point P when t = 0.9. (Round to the nearest hundredth).

( , )

In general, what does θ equal as a function of t?

θ = t

By repeating the process you used to arrive at the coordinates of point P when t = 0.9, write parametric equations for x and y as functions of t.

How many revolutions of t are needed to generate the entire graphs?

Diagram can be found blondebeliever.tumblr.[com]/precalc (on my blog) under question 3!

- Precalculus -
**Steve**, Friday, January 20, 2012 at 11:46pm
Let r be the radius of the small inside circle

Let R be the radius of the large outside circle

Let C be the center of the large circle

Cx = (r+R)cos(t)

Cy = (r+R)sin(t)

a = rt

A = a/R

θ = t+A

Px = Cx + Rcosθ

Py = Cy + Rsinθ

r = 2

R = 4

when t = 0.9

Cx = 6cos.9 = 3.73

Cy = 6sin.9 = 4.70

a = 2t = 1.80

A = 1.8/4 = 0.45

θ = t+A = 1.35

Px = 3.73 + 4cos1.35 = 4.61

Py = 4.70 + 4sin1.35 = 8.60

Px = 6cost + 4cos3t/2

Py = 6sint + 4sin3t/2

After t has gone once around, the outer circle has only made a half turn. So, after 2 turns of t, we have 3 turns of A.

- Precalculus -
**Steve**, Friday, January 20, 2012 at 11:47pm
fooplot has a good app for x-y plots, parametric plots, and polar plots

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