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Sketch the trochoid for the cases and .drdryrdcos xrdsin P0xdrPrdP2ye2txetysec2txcos tyt2xtye2txe3tyt4xt6yt2xt30yx238402yx2a3x2a2y2b2134.0, 32, 12, 1x2y12433.C1, 5B4, 2A1, 13, 12, 7P2x2, y2P1x1, y10t1yy1y2y1txx1x2x1t31.628||||CHAPTER 10PARAMETRIC EQUATIONS AND POLAR COORDINATES
LABORATORY PROJECTRUNNING CIRCLES AROUND CIRCLES||||629given by the parametric equationswhere is the acceleration due to gravity (m s ).(a) If a gun is fired with and ms, when will the bullet hit the ground? How far from the gun willit hit the ground? What is the maximum height reached by the bullet?;(b) Use a graphing device to check your answers to part (a).Then graph the path of the projectile for several other values of the angle to see where it hits the ground.Summarize your findings.(c) Show that the path is parabolic by eliminating the parameter.;Investigate the family of curves defined by the parametricequations , . How does the shape change as increases? Illustrate by graphing several members of thefamily.;48.The swallowtail catastrophe curvesare defined by the para-metric equations , . Graph several of these curves. What features do the curves have in common? How do they change when increases?50.Investigate the family of curves defined by the parametricequations , , where . Start by letting be a positive integer and see what happens to theshape as increases. Then explore some of the possibilitiesthat occur when is a fraction.cccc0ysin tsin ctxcos tnnbaybcos txasin nt49.cyct23t4x2ct4t3cyt3ctxt247.v05003029.8tyv0sin t12tt2xv0cos t(b) Use the geometric description of the curve to draw arough sketch of the curve by hand. Check your work byusing the parametric equations to graph the curve.;45.Suppose that the position of one particle at time is given byand the position of a second particle is given by(a) Graph the paths of both particles. How many points ofintersection are there?(b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place atthe same time? If so, find the collision points.(c) Describe what happens if the path of the second particleis given by46.If a projectile is fired with an initial velocity of meters persecond at an angle above the horizontal and air resistanceis assumed to be negligible, then its position after seconds is tv0x23cos ty21sin t0t20t2y21sin tx23cos t0t2y12 cos tx13 sin ttxOyAPx=2aBaIn this project we investigate families of curves, called hypocycloidsand epicycloids,that aregenerated by the motion of a point on a circle that rolls inside or outside another circle.