Chapter Review 751 In Problems 24–26, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. 24. x xy y 2 5 2 9 2 0 2 2 + + − = 25. x xy y 6 4 9 20 0 2 2 + + − = 26. x xy y x y 4 12 9 12 8 0 2 2 − + + + = In Problems 27–29, identify the conic that each polar equation represents, and graph it by hand.Verify your graph using a graphing utility. 27. r 4 1 cosθ = − 28. r 6 2 sinθ = − 29. r 8 4 8cosθ = + In Problems 30 and 31, convert each polar equation to a rectangular equation. 30. r 4 1 cosθ = − 31. r 8 4 8cosθ = + In Problems 32–34, graph the plane curve whose parametric equations are given, and show its orientation. Find a rectangular equation of each curve. Verify your graph using a graphing utility. 32. x t t y t t t 4 2, 1 ; ( ) ( ) = − = − −∞< <∞ 33. x t t y t t t 3sin , 4 cos 2; 0 2π ( ) ( ) = = + ≤ ≤ 34. x t t y t t t sec , tan ; 0 4 2 2 π ( ) ( ) = = ≤ ≤ 35. Find two different pairs of parametric equations for y x2 4. = − + 36. Find parametric equations for an object that moves along the ellipse x y 16 9 1, 2 2 + = where the motion begins at 4, 0 , ( ) is counterclockwise, and requires 4 seconds for a complete revolution. 37. Find an equation of the hyperbola whose foci are the vertices of the ellipse x y 4 9 36 2 2 + = and whose vertices are the foci of this ellipse. 38. Describe the collection of points in a plane so that the distance from each point to the point 3, 0 ( ) is three-fourths of its distance from the line x 16 3 . = 39. Searchlight A searchlight is shaped like a paraboloid of revolution. If a light source is located 1 foot from the vertex along the axis of symmetry and the opening is 2 feet across, how deep should the mirror be in order to reflect the light rays parallel to the axis of symmetry? 40. Semielliptical Arch Bridge A bridge is built in the shape of a semielliptical arch. The bridge has a span of 60 feet and a maximum height of 20 feet. Find the height of the arch at distances of 5, 10, and 20 feet from the center. 41. Calibrating Instruments In a test of their recording devices, a team of seismologists positioned two devices 2000 feet apart, with the device at point A to the west of the device at point B. At a point between the devices and 200 feet from point B, a small amount of explosive was detonated and a note made of the time at which the sound reached each device. A second explosion is to be carried out at a point directly north of point B. How far north should the site of the second explosion be chosen so that the measured time difference recorded by the devices for the second detonation is the same as that recorded for the first detonation? 42. Uniform Motion Mary’s train leaves at 7:15 am and accelerates at the rate of 3 meters per second per second. Mary, who can run 6 meters per second, arrives at the train station 2 seconds after the train has left. (a) Find parametric equations that model the motion of the train and Mary as a function of time. [Hint: The position s at time t of an object having acceleration a is s at 1 2 .2 = ] (b) Determine algebraically whether Mary will catch the train. If so, when? (c) Simulate the motions of the train and Mary by simultaneously graphing the equations found in part (a). 43. Projectile Motion Nick Foles throws a football with an initial speed of 80 feet per second at an angle of 35° to the horizontal. The ball leaves his hand at a height of 6 feet. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long is the ball in the air? (c) When is the ball at its maximum height? Determine the maximum height of the ball. (d) Determine the horizontal distance that the ball travels. (e) Using a graphing utility, simultaneously graph the equations found in part (a). 44. Formulate a strategy for discussing and graphing an equation of the form Ax Bxy Cy Dx Ey F 0 2 2 + + + + + =
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