492 CHAPTER 7 Analytic Trigonometry r h u Applications and Extensions Problems 83 and 84 require the following discussion: When granular materials are allowed to fall freely, they form conical (coneshaped) piles. The naturally occurring angle, measured from the horizontal, at which the loose material comes to rest is called the angle of repose and varies for different materials. The angle of repose θ is related to the height h and the base radius r of the conical pile by the equation r h cot . 1 θ = − See the figure . 83. Angle of Repose: De-icing Salt Due to potential transportation issues (for example, frozen waterways), de-icing salt used by highway departments in the Midwest must be ordered early and stored for future use.When de-icing salt is stored in a pile 14 feet high, the diameter of the base of the pile is 45 feet. (a) Find the angle of repose for de-icing salt. (b) What is the base diameter of a pile that is 17 feet high? (c) What is the height of a pile that has a base diameter of approximately 122 feet? Source: Salt Institute, The Salt Storage Handbook, 2015 84. Angle of Repose: Bunker Sand The steepness of sand bunkers on a golf course is affected by the angle of repose of the sand (a larger angle of repose allows for steeper bunkers). A freestanding pile of loose sand from a United States Golf Association (USGA) bunker had a height of 4 feet and a base diameter of approximately 6.68 feet. (a) Find the angle of repose for USGA bunker sand. (b) What is the height of such a pile if the diameter of the base is 8 feet? (c) A 6-foot-high pile of loose Tour Grade 50/50 sand has a base diameter of approximately 8.44 feet. Which type of sand (USGA or Tour Grade 50/50) would be better suited for steep bunkers? Source: Purdue University Turfgrass Science Program 85. Artillery A projectile fired into the first quadrant from the origin of a rectangular coordinate system will pass through the point x y , ( ) at time t according to the relationship x y gt cot 2 2 , 2 θ = + where the θ = angle of elevation of the launcher and g the = acceleration due to = gravity 32.2 feet second . 2 An artilleryman is firing at an enemy bunker located 2450 feet up the side of a hill that is 6175 feet away. He fires a round, and exactly 2.27 seconds later he scores a direct hit. (a) What angle of elevation did he use? (b) If the angle of elevation is also given by θ = v t x sec , 0 where v0 is the muzzle velocity of the weapon, find the muzzle velocity of the artillery piece he used. Source: www.egwald.com/geometry/projectile3d.php 86. Challenge Problem Find the exact value: cot sec sin 3 tan 6 1 π π ( ) + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − 87. Challenge Problem Write as an algebraic expression in x: x sec tan sin cos 1 1 ( ) [ ] { } − − Explaining Concepts 88. Explain in your own words how you would use your calculator to find the value of cot 10. 1− 89. Consult three books on calculus, and then write down the definition in each of y x sec 1 = − and y x csc . 1 = − Compare these with the definitions given in this text. Problems 90–99 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. Retain Your Knowledge 90. Find the complex zeros of f x x x ( ) 21 100. 4 2 = + − 91. Determine algebraically whether f x x x x ( ) 3 2 = + − is even, odd, or neither. 92. Convert 315° to radians. 93. Find the length of the arc subtended by a central angle of 75° on a circle of radius 6 inches. Give both the exact length and an approximation rounded to two decimal places. 94. Consider f x x x ( ) 2 10 3. 2 = − − + (a) Find the vertex. (b) Is the parabola concave up or concave down? (c) Find where f is increasing and where f is decreasing. 95. Solve: x log 16 2 5 2 ( ) + = 96. If f x x ( ) 3 = − and g x x x ( ) 7 4 = − − , find the domain of f g x . ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 97. Find the equation of a sine function with amplitude 4, period 3 , π and phase shift 1. 98. Rationalize the numerator: x c x c 1 1 2 2 − − − − 99. Find the average rate of change of f x x ( ) sin = from 2 π to 4 3 . π ‘Are You Prepared?’ Answers 1. Domain: x x odd integer multiples of 2 ; π { } ≠ Range: y y y 1 or 1 { } ≤− ≥ 2. True 3. 5 5
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