a. Parametric Line Problem: As you travel eastward on Highway I-90 from Cleveland to

Chapter 13, Problem R5

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a. Parametric Line Problem: As you travel eastward on Highway I-90 from Cleveland to Erie, the highway makes an angle of 24 north of east. Suppose you start at Cleveland at time t = 0 hours and drive 60 mi/hr. Write parametric equations for your position P(x, y) as functions of t hours, where x and y are in miles east and north of Cleveland, respectively. The highway passes through Erie, which is 50 miles east of Cleveland. How far north of Cleveland is Erie? b. Quarter and Dime Epicycloid Problem: A quarter is placed with its center at the origin of a coordinate system. A dime is placed with its center on the x-axis, as shown in Figure 13-6b. The quarter is held fixed while the dime rotates counterclockwise around it without slipping. Figure 13-6c shows the dime rotated to the place where a line from the origin through its center makes an angle of t radians with the x-axis. Point P on the dime traces an epicycloid. The quarter has radius 12 mm, and the dime has radius 9 mm. Figure 13-6b i. Do you agree that the circles shown in Figure 13-6b really are the size of a quarter and dime? Do you agree that the radii are 12 mm and 9 mm, respectively? Figure 13-6c ii. Derive a vector equation for point P(x, y) on the edge of the rolling dime in terms of t. To help you do this, notice that the position vector to P (not shown in Figure 13-6c) is equal to + , where extends from the origin to the center of the dime and extends from the center of the dime to the point P. Once you find in terms of in Figure 13-6c, you can get = by realizing that arc a on the quarter equals arc a on the dime, thus letting you get in terms of t. iii. Plot one complete cycle of the epicycloid with your grapher in parametric mode. Note that it takes more than one revolution for the graph to close.