SECTION 8.3 The Law of Cosines 575 A B C 49. Major League Baseball Field A major league baseball diamond is actually a square 90 feet on a side. The pitching rubber is located 60.5 feet from home plate on a line joining home plate and second base. (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base? 50. Little League Baseball Field According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base. (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base? 51. Finding the Length of a Guy Wire The height of a radio tower is 500 feet, and the ground on one side of the tower slopes upward at an angle of 10° (see the figure). (a) How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 100 feet from the base of the tower? (b) How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a position 100 feet from the base on the flat side? 52. Finding the Length of a Guy Wire A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of 5 .° See the figure. How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower? 53. Identifying Remains The Purkait triangle, located at the proximal end of the femur, has been used to identify the sex of fragmented skeletal remains. See the figure. (a) Given AB 30.1 mm, = AC 51.4 mm, = and A 89.2 , = ° find the length of BC. (b) If the average length of BC is 59.4 mm for men and 53.3 mm for women, which sex would be identified for the measurements in part (a)? 100 ft 500 ft 100 ft 108 100 ft 100 ft 58 500 ft A B C 54. Identifying Remains Like the Purkait triangle in Problem 53, the metric triangle is located at the proximal end of the femur and has been used to identify the sex of fragmented skeletal remains. See the figure. (a) If AC 48.8 mm, = BC 62.2 mm, = and C 89 , = ° find the length of AB. (b) If AB 80 < mm typically indicates a woman and AB 80 mm > typically indicates a man, which sex, if any, would be identified from the measurements in part (a)? 55. Soccer Angles A soccer goal is 8 yards wide. Suppose a goalie is standing on her line in the center of her goal as a striker from the opposing team moves the ball towards her. The near post angle, ,α is formed by rays extending from the ball to the near post and the goalie. Similarly, the far post angle, ,β is formed by rays extending from the ball to the far post and the goalie. See the figure. 8 yd Near post Goalie Far post a b (a) Determine the near post angle and the far post angle when the striker is 20 yards from the near post and 24 yards from the far post. (b) How far is the goalie from the ball? (c) To cover the near post, the goalie moves toward the near post to make the near post angle and the far post angle equal. How far toward her near post does the goalie need to move? 56. Covering the Angles In soccer, a defending goalkeeper wants to take up a position which bisects the angle that needs to be covered. See the figure. The keeper stands square to the ball—that is, perpendicular to the line of bisection—at a point where the area covered (shaded) lies completely outside the goal. How far is the goalkeeper from the center of the goal line if an attacking striker is 24 yards from the near post and 30 yards from the far post? d u u 8 yd 30 yd 24 yd
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