SECTION 7.3 Trigonometric Equations 501 of intersection of these two graphs. Now find the first two positive solutions of + = x x tan 0 rounded to two decimal places. 112. Carrying a Ladder around a Corner Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. See the figure. It can be shown that the length L of the ladder as a function of θ is θ θ θ ( ) = + L 4csc 3sec . u 4 ft 3 ft L (a) In calculus, you will be asked to find the length of the longest ladder that can turn the corner by solving the equation θ θ θ θ θ − = ° < < ° 3 sec tan 4 csc cot 0 0 90 Solve this equation for θ. (b) What is the length of the longest ladder that can be carried around the corner? (c) Graph θ θ ( ) = ° ≤ ≤ ° L L , 0 90 , and find the angle θ that minimizes the length L. (d) Compare the result with the one found in part (a). Explain why the two answers are the same. 113. Projectile Motion The horizontal distance that a projectile will travel in the air (ignoring air resistance) is given by the equation θ θ ( ) ( ) = R v g sin 2 0 2 where v0 is the initial velocity of the projectile, θ is the angle of elevation, and g is acceleration due to gravity (9.8 meters per second squared). (a) If you can throw a baseball with an initial speed of 34.8 meters per second, at what angle of elevation θ should you direct the throw so that the ball travels a distance of 107 meters before striking the ground? (b) Determine the maximum distance that you can throw the ball. (c) Graph θ( ) = R R , with = v 34.8 0 meters per second. (d) Verify the results obtained in parts (a) and (b) using a graphing utility. 114. Projectile Motion Refer to Problem 113. (a) If you can throw a baseball with an initial speed of 40 meters per second, at what angle of elevation θ should you direct the throw so that the ball travels a distance of 110 meters before striking the ground? (b) Determine the maximum distance that you can throw the ball. (c) Graph θ( ) = R R , with = v 40 0 meters per second. (d) Verify the results obtained in parts (a) and (b) using a graphing utility. 107. Blood Pressure Several research papers use a sinusoidal graph to model blood pressure. Assuming that a person’s heart beats 70 times per minute, the blood pressure P of an individual after t seconds can be modeled by the function π ( ) ( ) = + P t t 20 sin 7 3 100 (a) In the interval [ ] 0, 1 , determine the times at which the blood pressure is 100 mm Hg. (b) In the interval [ ] 0, 1 , determine the times at which the blood pressure is 120 mm Hg. (c) In the interval [ ] 0, 1 , determine the times at which the blood pressure is between 100 and 105 mm Hg. 108. The Ferris Wheel In 1893, George Ferris engineered the Ferris wheel. It was 250 feet in diameter. If a Ferris wheel makes 1 revolution every 40 seconds, then the function π ( ) ( ) = − + h t t 125 sin 0.157 2 125 represents the height h, in feet, of a seat on the wheel as a function of time t, where t is measured in seconds. The ride begins when = t 0. (a) During the first 40 seconds of the ride, at what time t is an individual on the Ferris wheel exactly 125 feet above the ground? (b) During the first 80 seconds of the ride, at what time t is an individual on the Ferris wheel exactly 250 feet above the ground? (c) During the first 40 seconds of the ride, over what interval of time t is an individual on the Ferris wheel more than 125 feet above the ground? 109. Holding Pattern An airplane is asked to stay within a holding pattern near Chicago’s O’Hare International Airport. The function ( ) ( ) = + d x x 70sin 0.65 150 represents the distance d, in miles, of the airplane from the airport at time x, in minutes. (a) When the plane enters the holding pattern, = x 0, how far is it from O’Hare? (b) During the first 20 minutes after the plane enters the holding pattern, at what time x is the plane exactly 100 miles from the airport? (c) During the first 20 minutes after the plane enters the holding pattern, at what time x is the plane more than 100 miles from the airport? (d) While the plane is in the holding pattern, will it ever be within 70 miles of the airport? Why? 110. Projectile Motion A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range R of the ball as a function of the angle θ to the horizontal is given by θ θ ( ) ( ) = R 672 sin 2 , where R is measured in feet. (a) At what angle θ should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)? (b) At what angle θ should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)? (c) At what angle θ should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)? (d) Can the golfer hit the ball 720 feet (240 yards)? 111. Heat Transfer In the study of heat transfer, the equation + = x x tan 0 occurs. Graph = − Y x 1 and = Y x tan 2 for ≥ x 0. Conclude that there are an infinite number of points Applications and Extensions
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