410 CHAPTER 6 Trigonometric Functions For Problems 121–124, use the following discussion. Projectile Motion The path of a projectile fired at an inclination θ to the horizontal with initial speed v0 is a parabola (see the figure). Range, R Height, H v0 = Initial speed u The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the function R v g sin 2 0 2 θ θ ( ) ( ) = where g 32.2 ≈ feet per second per second 9.8 ≈ meters per second per second is the acceleration due to gravity. The maximum height H of the projectile is given by the function H v g sin 2 0 2 2 θ θ ( ) ( ) = In Problems 121–124, find the range R and maximum height H. Round answers to two decimal places. See the discussion above. 121. The projectile is fired at an angle of 45° to the horizontal with an initial speed of 100 feet per second. 122. The projectile is fired at an angle of 30° to the horizontal with an initial speed of 150 meters per second. 123. The projectile is fired at an angle of 25° to the horizontal with an initial speed of 500 meters per second. 124. The projectile is fired at an angle of 50° to the horizontal with an initial speed of 200 feet per second. 125. Inclined Plane See the figure. If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the function t a g 2 sin cos θ θ θ ( ) = where a is the length (in feet) of the base and g 32 feet ≈ per second per second is the acceleration due to gravity. How long does it take a block to slide down an inclined plane with base a 10 feet = when: (a) 30 ? θ = ° (b) 45 ? θ = ° (c) 60 ? θ = ° u a 120. Use a calculator in radian mode to complete the following table. What do you conjecture about the value of g cos 1 θ θ θ ( ) = − as θ approaches 0? θ 0.5 0.4 0.2 0.1 0.01 0.001 0.0001 0.00001 cos 1 θ − g cos 1 θ θ θ ( ) = − 126. Piston Engines In a certain piston engine, the distance x (in centimeters) from the center of the drive shaft to the head of the piston is given by the function x cos 16 0.5 cos 2 θ θ θ ( ) ( ) = + + where θ is the angle between the crank and the path of the piston head. See the figure. Find x when 30 θ = ° and when 45 θ = ° x U 127. Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. See the figure. Ocean Beach Paved path River 1 mi x 4 mi 4 mi u u Sally can jog 8 miles per hour along the paved path, but only 3 miles per hour in the sand on the beach. Because of a river directly between the two houses, it is necessary to jog in the sand to the path, continue on the path, and then jog directly back in the sand to get from one house to the other. The time T to get from one house to the other as a function of the angle θ shown in the figure is T 1 2 3sin 1 4tan , 0 90 θ θ θ θ ( ) = + − ° < < ° (a) Calculate the time T for 30 . θ = ° How long is Sally on the paved path? (b) Calculate the time T for 45 . θ = ° How long is Sally on the paved path? (c) Calculate the time T for 60 . θ = ° How long is Sally on the paved path? (d) Calculate the time T for 90 . θ = ° Describe the route taken. Why can’t the formula for T be used? 128. Designing Fine Decorative Pieces A designer of decorative art plans to market solid gold spheres encased in clear crystal cones. Each sphere is of fixed radius R and will be enclosed in
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