859 8.8 Parametric Equations, Graphs, and Applications 49. Flight of a Softball Sally hits a softball when it is 2 ft above the ground. The ball leaves her bat at an angle of 20° with respect to the ground at velocity 88 ft per sec. 50. Flight of a Baseball Francisco hits a baseball when it is 2.5 ft above the ground. The ball leaves his bat at an angle of 29° from the horizontal with velocity 136 ft per sec. 298 2.5 ft (Modeling) Solve each problem. See Examples 7 and 8. 51. Path of a Rocket A rocket is launched from the top of an 8-ft platform. Its initial velocity is 128 ft per sec. It is launched at an angle of 60° with respect to the ground. (a) Find the rectangular equation that models its path. What type of path does the rocket follow? (b) Determine the total flight time, to the nearest second, and the horizontal distance the rocket travels, to the nearest foot. 52. Simulating Gravity on the Moon If an object is thrown on the moon, then the parametric equations of flight are x = 1v cos u2t and y = 1v sin u2t - 2.66t2 + h. Find the distance, to the nearest foot, a golf ball hit at 88 ft per sec (60 mph) at an angle of 45° with the horizontal travels on the moon if the moon’s surface is level. 53. Flight of a Baseball A baseball is hit from a height of 3 ft at a 60° angle above the horizontal. Its initial velocity is 64 ft per sec. (a) Write parametric equations that model the flight of the baseball. (b) Determine the horizontal distance, to the nearest tenth of a foot, traveled by the ball in the air. Assume that the ground is level. (c) What is the maximum height of the baseball, to the nearest tenth of a foot? At that time, how far has the ball traveled horizontally? (d) Would the ball clear a 5-ft-high fence that is 100 ft from the batter? 54. Path of a Projectile A projectile has been launched from the ground with initial velocity 88 ft per sec. The parametric equations x = 82.7t and y = -16t2 + 30.1t model the path of the projectile, where t is in seconds. (a) Determine the angle u, to the nearest tenth of a degree, that the projectile makes with the horizontal at the launch. (b) Write parametric equations for the path using the cosine and sine functions. Work each problem. 55. Give two parametric representations of the parabola y = a1x - h22 + k. 56. Give a parametric representation of the rectangular equation x 2 a2 - y2 b2 = 1. 57. Give a parametric representation of the rectangular equation x 2 a2 + y2 b2 = 1. 58. The spiral of Archimedes has polar equation r = a u, where r2 = x2 + y2. Show that a parametric representation of the spiral of Archimedes is x = au cos u, y = au sin u, for u in 1-∞, ∞2.
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