SECTION 3.5 Inequalities Involving Quadratic Functions 183 (b) If the goal in part (a) is to hit a target on the ground 75 kilometers away, is it possible to do so? If so, for what values of c? If not, what is the maximum distance the round will travel? Source: www.answers.com 38. Challenge Problem Runaway Car Using Hooke’s Law, we can show that the work W done in compressing a spring a distance of x feet from its at-rest position is W kx 1 2 ,2 = where k is a stiffness constant depending on the spring. It can also be shown that the work done by a body in motion before it comes to rest is given by W w g v 2 ,2 = where w g weight of the object in lb , acceleration due to ( ) = = gravity 32.2 ft s , 2 ( ) and v object’s velocity in ft s . ( ) = A parking garage has a spring shock absorber at the end of a ramp to stop runaway cars. The spring has a stiffness constant k 9450 lb ft = and must be able to stop a 4000-lb car traveling at 25 mph. What is the least compression required of the spring? Express your answer using feet to the nearest tenth. Source: www.sciforums.com 36. Revenue The John Deere company has found that the revenue from sales of heavy-duty tractors is a function of the unit price p, in dollars, that it charges. The revenue R, in dollars, is given by R p p p 1 2 1900 2 ( ) = − + (a) At what prices p is revenue zero? (b) For what range of prices will revenue exceed $1,200,000? 37. Artillery A projectile fired from the point 0, 0 ( ) at an angle to the positive x-axis has a trajectory given by y cx c g x v 1 2 2 2 ( )( ) ( ) = − + where x y v g c horizontal distance in meters height in meters initial muzzle velocity in meters per second m s acceleration due to gravity 9.81 meters per second squared ms 0 is a constant determined by the angle of elevation. 2 ( ) ( ) = = = = = > A howitzer fires an artillery round with a muzzle velocity of 897 m s. (a) If the round must clear a hill 200 meters high at a distance of 2000 meters in front of the howitzer, what c values are permitted in the trajectory equation? Explaining Concepts 39. Show that the inequality x 4 0 2 ( ) − ≤ has exactly one solution. 40. Show that the inequality x 2 0 2 ( ) − > has one real number that is not a solution. 41. Explain why the inequality x x 1 0 2 + + > has all real numbers as the solution set. 42. Explain why the inequality x x 1 0 2 − + < has the empty set as the solution set. 43. Explain the circumstances under which the x-intercepts of the graph of a quadratic function are included in the solution set of a quadratic inequality. Retain Your Knowledge Problems 44–53 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. 44. Find the domain of f x x 10 2 . ( ) = − 45. Determine algebraically whether f x x x 9 2 ( ) = − + is even, odd, or neither. 46. Suppose f x x 2 3 6. ( ) = − (a) Find the intercepts of the graph of f. (b) Graph f. 47. Write a general formula to describe the variation: d varies directly with t; d 203 = when t 3.5 = . 48. Find the zeros of f x x x6 8 2 ( ) = + − . 49. Find the intercepts of the graph of y x x 4 25 1 . 2 2 = − − In Problems 50 and 51, if f x x x2 7 2 ( ) = + − and g x x3 4, ( ) = − find: 50. g f x ( )( ) − 51. f g x ( )( ) ⋅ 52. Find the difference quotient of f f x x x : 3 5 2 ( ) = − 53. Simplify: x x x x x 5 2 7 8 2 7 2 7 4 4 5 3 8 ( ) ( ) ( ) + − + + ‘Are You Prepared?’ Answers 1. x x 3 { } >− or 3, ( ) − ∞ 2. x 2 7 − < ≤
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