748 CHAPTER 10 Analytic Geometry 58. Uniform Motion A Cessna (heading south at 120 mph) and a Boeing 737 (heading west at 600 mph) are flying toward the same point at the same altitude. The Cessna is 100 miles from the point where the flight patterns intersect, and the 737 is 550 miles from this intersection point. See the figure. 550 mi 100 mi 120 mph 600 mph N S E W (a) Find parametric equations that model the motion of the Cessna and the 737. (b) Find a formula for the distance between the planes as a function of time. (c) Graph the function in part (b) using a graphing utility. (d) What is the minimum distance between the planes? When are the planes closest? (e) Simulate the motion of the planes by simultaneously graphing the equations found in part (a). 59. The Green Monster The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet above the ground, at an angle of 45 .° Use g 32 ft sec2 = as the acceleration due to gravity, and ignore any air resistance. (a) Find parametric equations that model the position of the ball as a function of time. (b) What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet. (c) How far is the ball from home plate at its maximum height? Give your answer in feet. (d) If the ball is hit straight down the left field line, will it clear the Green Monster? If it does, by how much does it clear the wall? Source: The Boston Red Sox 60. Projectile Motion The position of a projectile fired with an initial velocity v0 feet per second and at an angle θ to the horizontal at the end of t seconds is given by the parametric equations x t v t y t v t t cos sin 16 0 0 2 θ θ ( ) ( ) ( ) ( ) = = − See the figure (top right). R u (a) Obtain a rectangular equation of the trajectory, and identify the curve. (b) Show that the projectile hits the ground y 0 ( ) = when t v 1 16 sin . 0 θ = (c) How far has the projectile traveled (horizontally) when it strikes the ground? In other words, find the range R. (d) Find the time t when x y. = Next find the horizontal distance x and the vertical distance y traveled by the projectile in this time. Then compute x y . 2 2 + This is the distance R, the range, that the projectile travels up a plane inclined at 45° to the horizontal x y . ( ) = See the following figure. (See also Problem 105 in Section 7.6.) R 458 u 61. Show that parametric equations for a line passing through the points x y , 1 1 ( ) and x y , 2 2 ( ) are x t x x t x y t y y t y t 2 1 1 2 1 1 ( ) ( ) ( ) ( ) = − + = − + −∞< <∞ What is the orientation of this line? 62. Hypocycloid The hypocycloid is a plane curve defined by the parametric equations x t t y t t t cos sin 0 2 3 3 π ( ) ( ) = = ≤ ≤ (a) Graph the hypocycloid using a graphing utility. (b) Find a rectangular equation of the hypocycloid. 63. Challenge Problem Find parametric equations for the circle x y R , 2 2 2 + = using as the parameter the slope m of the line through the point R, 0 ( ) − and a general point P x y , ( ) = on the circle. 64. Challenge Problem Find parametric equations for the parabola y x ,2 = using as the parameter the slope m of the line joining the point 1, 1 ( ) to a general point P x y , ( ) = of the parabola. Explaining Concepts 65. In Problem 62, we graphed the hypocycloid. Now graph the rectangular equations of the hypocycloid. Did you obtain a complete graph? If not, experiment until you do. 66. Research plane curves called hypocycloid and epicycloid . Write a report on what you find. Compare and contrast them to a cycloid.
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