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Precalculus

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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 -

    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 -

    fooplot has a good app for x-y plots, parametric plots, and polar plots

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