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# Epicycloid Problem: Figure 13-5l (top diagram) shows the epicycloid traced by a point on

ISBN: 9781559533911 468

## Solution for problem 8 Chapter 13-5

Precalculus with Trigonometry: Concepts and Applications | 1st Edition

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Problem 8

Epicycloid Problem: Figure 13-5l (top diagram) shows the epicycloid traced by a point on the rim of a wheel of radius 2 cm as it rotates, without slipping, around the outside of a circle of radius 6 cm. The wheel starts with point P(x, y) at (6, 0). The parameter t is the number of radians from the positive x-axis to a line through the center of the wheel. Let (not shown) be the position vector to point P(x, y). Let be the vector from the origin to the center of the wheel. Let be the vector from the center of the wheel to point P(x, y). Figure 13-5l a. Find in terms of t and the unit vectors and . Find in terms of angle in Figure 13-5l and the unit vectors and . b. Write in terms of t by observing that starts at radians when t = 0. Thus, is given by the sum + A + B, where A and B are shown in the lower diagram of Figure 13-5l. Express angles A and B in terms of t. Note that the arc of the wheel subtended by angle B equals the arc of the circle subtended by angle t, because the wheel rotates without slipping. Note also that the length of an arc of a circle equals the central angle in radians times the radius. c. Write a vector equation for as a function of t by observing that . Use the result to plot the epicycloid using parametric mode. Use a t-range large enough to get one complete cycle, and use equal scales on both axes. Does the graph agree with Figure 13-5l? If not, go back and check your work. d. Plot the 6-cm circle on the same screen as in part c. Does the result agree with the figure?

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