66.Determineparametricequationsforthecurvesdefinedbyx2n+y2n=r2nfor integersn. (Hint: Start withn=1,x2+y2=r2, then think of the general equation as (xn)2+(yn)2=r2n.) Sketch the graphs forn=1,n=2 andn=3,and predict what the curve will look like for large values ofn.◦and a 27-second delay. How close does the Flying Zucchiniget to the special seat? Given that a Ferris wheel seat actuallyhas height, width, and depth, do you think that this is closeenough? Repeat with (a) initial velocity 75 ft/s, launch angle47◦and 47.25-second delay; (b) initial velocity 118 ft/s, launchangle 35◦and 28-second delay. Develop criteria for a safe andexciting human cannonball act. Consider each of the following:Should the launch velocity be large or small? Should the seatbe high or low when the cannonball lands? Should the humanhave a positive or negative vertical velocity at landing? Howclose (vertically and horizontally) should the human need toget to the center of the seat? Based on your criteria, which ofthe launches in this exercise is the best? Find an initial velocity,launch angle and launch delay that is better.EXPLORATORY EXERCISES

Assume that the central arm has length 200 feet and rotatesabout its center. Also assume that the wheels have radius 40 feetand rotate at the same speed as the central arm. Find parametricequations for the position of a rider and graph the rider’s path.

2.The Flying Zucchini Circus Troupe has a human cannon-ball act, shooting a performer from a cannon into a spe-cially padded seat of a turning Ferris wheel. The Ferriswheel has a radius of 40 feet and rotates counterclockwise atone revolution per minute. The special seat starts at groundlevel. Carefully explain why parametric equations for theseat arex=40 cos(π30t−π2)

30◦and a 27-second delay. How close does the Flying Zucchiniget to the special seat? Given that a Ferris wheel seat actuallyhas height, width, and depth, do you think that this is closeenough? Repeat with (a) initial velocity 75 ft/s, launch angle47◦and 47.25-second delay; (b) initial velocity 118 ft/s, launchangle 35◦and 28-second delay. Develop criteria for a safe andexciting human cannonball act. Consider each of the following:Should the launch velocity be large or small? Should the seatbe high or low when the cannonball lands? Should the humanhave a positive or negative vertical velocity at landing? Howclose (vertically and horizontally) should the human need toget to the center of the seat? Based on your criteria, which ofthe launches in this exercise is the best? Find an initial velocity,launch angle and launch delay that is better.

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