SECTION 10.7 Plane Curves and Parametric Equations 735 49. Challenge Problem Board Deflection A crate is placed at the center of a 5-meter board that is supported only at its ends.The weight of the crate causes a deflection of the board at its center. If the shape of the deflected board is a parabola given by r 250 1 sinθ = − determine the amount of deflection at the center assuming the focus is at the pole. Displacement (Not to scale) 5 m 2.5 m 50. Challenge Problem Suppose that a conic has an equation of the form r ep e 1 sin . θ = − If the polar coordinates of two points on the graph are M, 2 π ( ) and m, 3 2 , π ( ) show that e M m M m p mM M m and 2 = − + = − 51. Challenge Problem Escape Velocity From physics, the equation for the free-flight trajectory of a satellite launched a distance r0 from the center of the earth is given by the polar equation r r GM r v GM r v 1 1 1 cos e e 0 0 0 2 0 2 0 2 θ = − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + where Me is the mass of the earth, G is the gravitational constant, and v0 is the initial velocity of the satellite. If the initial velocity is equal to the escape velocity (the velocity needed to overcome Earth’s gravitational pull) then the resulting trajectory follows a parabolic path. What is the escape velocity? Retain Your Knowledge Problems 52–61 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. 52. Find the area of the triangle described: a b 7, 8, = = and c 10. = Round to two decimal places. 53. Without graphing, determine the amplitude and period of y x 4cos 1 5 . ( ) = 54. Solve: x x 2 cos cos 1 0, 2 + − = x 0 2π ≤ < 55. For v i j 10 24 , = − find v . 56. If an arc length of 14 feet subtends a central angle of 105°, what is the radius of the circle? 57. A radioactive substance has a half-life of 15 years. How long until there is 40% of a sample remaining? 58. Determine where the function f x x x x x 3 if 2 1 1 if 1 2 ( ) = + − ≤ <− + ≥− ⎧ ⎨ ⎪⎪ ⎩ ⎪⎪ is increasing, decreasing, and constant. 59. Find k so that y kx sin( ) = has a period of 5 6 . π 60. Write the vertex form of the quadratic function whose graph has vertex 3, 8 ( ) − and y -intercept 5. 61. Find the area of the region bounded by the graph of f x x 1 2 3, ( ) = + the x -axis, and the vertical lines x 0 = and x 8. = ‘Are You Prepared?’ Answers 1. r r cos; sin θ θ 2. x y x x y 6 or 3 9 2 2 2 2 ( ) + = − + = OBJECTIVES 1 Graph Parametric Equations by Hand (p. 736) 2 Graph Parametric Equations Using a Graphing Utility (p. 737) 3 Find a Rectangular Equation for a Plane Curve Defined Parametrically (p. 738) 4 Use Time as a Parameter in Parametric Equations (p. 740) 5 Find Parametric Equations for Plane Curves Defined by Rectangular Equations (p. 743) 10.7 Plane Curves and Parametric Equations • Amplitude and Period of Sinusoidal Graphs (Section 6.4, pp. 430–432) Now Work the ‘Are You Prepared?’ problem on page 745. PREPARING FOR THIS SECTION Before getting started, review the following:
RkJQdWJsaXNoZXIy NjM5ODQ=