SECTION 10.7 Plane Curves and Parametric Equations 747 (b) How long was the ball in the air? (c) Determine the horizontal distance that the ball traveled. (d) When was the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a). 55. Projectile Motion Suppose a golf ball is hit off a cliff 300 meters high with an initial speed of 40 meters per second at an angle of 45° to the horizontal. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long is the ball in the air? (c) Determine the horizontal distance that the ball travels. (d) When is the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a). 56. Projectile Motion Suppose a golf ball is hit on the moon off a cliff 300 meters high with an initial speed of 40 meters per second at an angle of 45° to the horizontal. Hint: Gravity on the Moon is one-sixth of that on Earth. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long is the ball in the air? (c) Determine the horizontal distance that the ball travels. (d) When is the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a). 57. Uniform Motion A Toyota Camry (traveling east at 40 mph) and a Chevy Impala (traveling north at 30 mph) are heading toward the same intersection. The Camry is 5 miles from the intersection when the Impala is 4 miles from the intersection. See the figure. DRIVE THRU N S E W 5 mi 40 mph 30 mph 4 mi (a) Find parametric equations that model the motion of the Camry and the Impala. (b) Find a formula for the distance between the cars as a function of time. (c) Graph the function in part (b) using a graphing utility. (d) What is the minimum distance between the cars? When are the cars closest? (e) Simulate the motion of the cars by simultaneously graphing the equations found in part (a). 49. Projectile Motion Bob throws a ball straight up with an initial speed of 50 feet per second from a height of 6 feet. (a) Find parametric equations that model the motion of the ball as a function of time. (b) How long is the ball in the air? (c) When is the ball at its maximum height? Determine the maximum height of the ball. (d) Simulate the motion of the ball by graphing the equations found in part (a). 50. Projectile Motion Alice throws a ball straight up with an initial speed of 40 feet per second from a height of 5 feet. (a) Find parametric equations that model the motion of the ball as a function of time. (b) How long is the ball in the air? (c) When is the ball at its maximum height? Determine the maximum height of the ball. (d) Simulate the motion of the ball by graphing the equations found in part (a). 51. Catching a Train Bill’s train leaves at 8:06 am and accelerates at the rate of 2 meters per second per second. Bill, who can run 5 meters per second, arrives at the train station 5 seconds after the train has left and runs for the train. (a) Find parametric equations that model the motions of the train and Bill as a function of time. [Hint: The position s at time t of an object having acceleration a is s at 1 2 .2 = ] (b) Determine algebraically whether Bill will catch the train. If so, when? (c) Simulate the motion of the train and Bill by simultaneously graphing the equations found in part (a). 52. Catching a Bus Jodi’s bus leaves at 5:30 pm and accelerates at the rate of 3 meters per second per second. Jodi, who can run 5 meters per second, arrives at the bus station 2 seconds after the bus has left and runs for the bus. (a) Find parametric equations that model the motions of the bus and Jodi as a function of time. [Hint: The position s at time t of an object having acceleration a is s at 1 2 .2 = ] (b) Determine algebraically whether Jodi will catch the bus. If so, when? (c) Simulate the motion of the bus and Jodi by graphing simultaneously the equations found in part (a). 53. Projectile Motion Sean throws a baseball with an initial speed of 145 feet per second at an angle of 20° to the horizontal. The ball leaves Sean’s hand at a height of 5 feet. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long is the ball in the air? (c) Determine the horizontal distance that the ball travels. (d) When is the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a). 54. Projectile Motion Billy hit a baseball with an initial speed of 125 feet per second at an angle of 40° to the horizontal. The ball was hit at a height of 3 feet above the ground. (a) Find parametric equations that model the position of the ball as a function of time. Applications and Extensions
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