Skills Practice - Rcboe.org

Skills Practice Skills Practice for Lesson 15.1Name _____________________________________________Date ____________________Name That ConicClassifying Equations of Conics in General FormProblem SetWrite the equation of each conic in standard form. Then, determine the typeof conic section represented by the equation.1. x 2 4x y 6 0( x 2 4x) y 6x 2 4x 4 y 6 4( x 2) 2 y 2( x 2) 2 ( y 2)This conic section is a parabola.2. x 2 y 2 10x 2y 4 0© 2010 Carnegie Learning, Inc.3. 8x 2 2y 2 16 015Chapter 15 l Skills Practice 947

4. 9x 2 4y 2 72x 80y 292 05. x 2 y 2 2 06. 7x 2 9y 2 14x 54y 38 0© 2010 Carnegie Learning, Inc.15948 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________7. x 2 y 2 12x 10 08. 5x 2 y 2 20x 14y 6 09. y 2 3x 0© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 949

10. 15x 2 12y 2 60 011. y 2 8x 2y 7 012. 3x 2 5y 2 36x 80y 23 0© 2010 Carnegie Learning, Inc.15950 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________Determine the type of conic section represented by each equation.13. 2 x 2 4y 6x 11 0parabola because B 0 and A 014. x 2 8y 2 2 x 3y 10 015. 4x 2 2y 2 12 x y 616. x 2 y 2 5x 7y 15 017. 16x 2 16y 2 128© 2010 Carnegie Learning, Inc.18. y 2 4x 019. 16x 2 16y 2 32x 64y 128 020. 2 x 2 3y 2 10x 32y 100 015Chapter 15 l Skills Practice 951

© 2010 Carnegie Learning, Inc.15952 Chapter 15 l Skills Practice

Skills Practice Skills Practice for Lesson 15.2Name _____________________________________________Date ____________________Pulling It All TogetherEquations and Graphs of ConicsProblem SetDetermine the type of conic section represented by each equation.1. 2 x 2 5y 2 3x 9y 6 0ellipse because A B, but A and B have the same signs2. 2 x 2 5y 2 3x 9y 6 03. 2 x 2 3x 9y 6 04. 2 x 2 2y 2 3x 9y 6 05. 4x 2 9y 2 12 06. 4x 9y 2 12 0© 2010 Carnegie Learning, Inc.Write each equation in standard form and determine the type of conic representedby the equation. Identify the center and radius for each circle. Identify the center,vertices, co-vertices, foci, and eccentricity for each ellipse. Identify the center,vertices, co-vertices, foci, asymptotes, and eccentricity for each hyperbola. Identifythe vertex, axis of symmetry, focus, directrix, and concavity for each parabola.7. x 2 y 2 6x 2y 6 0This equation represents a circle.x 2 6x y 2 2y 6( x 2 6x 9) ( y 2 2y 1) 6 9 1( x 3) 2 ( y 1) 2 415center is (3, 1) and radius 2Chapter 15 l Skills Practice 953

8. 16x 2 9y 2 128x 36y 76 09. y 2 16x 4y 52 0© 2010 Carnegie Learning, Inc.15954 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________10. 36x 2 49y 2 1764 011. x 2 20y 0© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 955

12. x 2 y 2 10y 24 013. 9x 2 4y 2 108x 56y 484 0© 2010 Carnegie Learning, Inc.15956 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________14. 4x 2 25y 2 250y 525 0© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 957

Graph each conic section.15. ( x 3) 2 ( y 2) 2 25 16. ( x 1) 2 4yyyxx(3, –2)17. ( _______ x 4)236 ( _______ y 5)2 1 18. 18x 2 8y 2 72 09yyxx© 2010 Carnegie Learning, Inc.15958 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________19. ( y 3) 2 12( x 4) 20. __ x24 ( _______ y 6)225y 1yxx© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 959

© 2010 Carnegie Learning, Inc.15960 Chapter 15 l Skills Practice

Skills Practice Skills Practice for Lesson 15.3Name _____________________________________________Date ____________________Applications of ConicsConic Sections and Problem SolvingProblem SetWrite an equation of a conic section to model each situation.1. The main cables of a suspension bridge are parabolic. The parabolic shape allowsthe cables to bear the weight of the bridge evenly. The distance between thetowers is 300 feet and the height of each tower is about 50 feet. Write an equationfor the parabola representing the cable between the two towers.300 ft50 ftPlace the bridge on a coordinate plane with the center of the bridge at the origin.Three coordinate pairs are known: (0, 0), (150, 50), and (150, 50).Since the vertex is (0, 0), the equation is of the form x 2 4py.Substitute (150, 50) into the equation and calculate 4p.150 2 4p(50)22,500 4p(50)© 2010 Carnegie Learning, Inc.4p 450The equation is x 2 450y.15Chapter 15 l Skills Practice 961

2. The cross section of a satellite dish is a parabola. The satellite dish is 8 feet wideat its opening and 1 foot deep. Write an equation for the parabola representing thesatellite dish.3. A planet's orbit is centered at the origin with the major axis of its orbit along thex-axis and with the sun to the left of the origin. For the ellipse modeling the orbitof this planet, the distance between the vertices is approximately 0.9 AU and thedistance between the foci is approximately 0.26 AU. Write an equation for theellipse modeling the orbit of this planet.© 2010 Carnegie Learning, Inc.15962 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________4. An elliptical whispering gallery has an inner chamber that is 130 feet long and60 feet high above eye level. The center of the room at eye level is located at theorigin of the ellipse. Write an equation to model the shape of the whispering gallery.5. Cooling towers for nuclear power plants are hyperbolic. The shape of these towersis formed by rotating a hyperbola around a vertical axis to form a three-dimensionalshape called a hyperboloid. Typically, these cooling towers are slightly larger onthe bottom than the top. The design of a hyperboloid creates a draft bringing coolair into the system to aid in the cooling process. Consider the hyperboloid shown.Each coordinate is measured in feet. Write an equation for the hyperbola used togenerate this three-dimensional cooling tower.© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 963

6. Plotted on a coordinate plane in which one unit equals one mile, the coordinatesof transmitter A are (0, 130), the coordinates of transmitter B are (0, 0), and thecoordinates of transmitter C are (126, 0). A ship is located 80 miles from transmitter A,56 miles from transmitter B, and 65 miles from transmitter C. Write an equation of thehyperbola with foci at transmitters A and B which would be used to help determinethe location of the ship.Use the equation of a conic section to solve each problem.7. Many amusement parks have mirrors that are parabolic.The focal length of a mirror is the distance from the vertexto the focus of the mirror. Consider a mirror that is 60 inchestall with a vertex that is concave 4 inches from the top and bottomedges of the mirror. Calculate the focal length of the mirror.60 in.Place the parabola that represents the mirror on the coordinateplane with its vertex at the origin. Three coordinate pairs areknown: (0, 0), (4, 30), and (4, 30). The standard form equationof a parabola opening to the right with vertex at theorigin is y 2 4px.Substitute (4, 30) into the equation.30 2 4p(4)900 4p(4)4 in.© 2010 Carnegie Learning, Inc.4p 22515p 56.25The focal length of the mirror is 56.25 inches.964 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________8. The surface of a flashlight can be represented by the equation y ___ 110 x2 .The dimensions are in inches. A point on the parabola that represents the surfaceof the flashlight is (20, 40). Let f represent the distance from the vertex to the focus.Determine the focus of the parabola.© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 965

9. The entrance to a zoo is an archway in the shape of half an ellipse. The archway is20 feet wide and 15 feet high in the middle. If a giraffe walks through, centered atpoint A, 5 feet from the edge of the archway, and the giraffe is 12 feet tall, will it fitstanding up or will it need to duck its neck down?15 ft20 ftA 5 ft10. The swimming pool at a resort is in the shape of an ellipse. The major axis of theellipse is 72 feet long. A fountain is located at one focus. The eccentricity of theellipse is __ 2 . If Henri sits on the edge of the pool, calculate the closest distance he3could be to the fountain and the furthest distance he could be from the fountain.© 2010 Carnegie Learning, Inc.15966 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________11. A floodlight’s surface can be represented by the equation ___ x264 ___y236 1.The dimensions are in inches. The light source must be placed at a focus of thehyperbola. Determine how far the light source should be placed from the vertexof the hyperbolic surface to create the floodlight.12. Radio signals submitted from a transmitter form a pattern of concentric circles.A radio station is located at the origin. Lois lives 24 miles north and 45 miles eastof the station. Her home is located on the edge of this station’s broadcast range.Write an equation of a circle to represent the maximum listening area. Mark lives 73 mileswest and 15 miles north of Lois. Can Mark listen to this radio station from his home?© 2010 Carnegie Learning, Inc.15Chapter 15 l Skills Practice 967

© 2010 Carnegie Learning, Inc.15968 Chapter 15 l Skills Practice

Skills Practice Skills Practice for Lesson 15.4Name _____________________________________________Date ____________________Spheres and PlanesIntersections of Spheres and PlanesVocabularyMatch each definition to its corresponding term.1. the distance from a point on the sphere a. sphereto its centerb. radius of a sphere2. x 2 y 2 z 2 r 2c. great circled. standard form of theequation of a sphere.3. a set of all points in three-dimensionalspace equidistant from a fixed point4. a cross section of a sphere and a planethat passes through the center of the sphere© 2010 Carnegie Learning, Inc.Problem SetWrite an equation in standard form of each sphere given its center and radius.1. center (1, 2, 3), radius 6( x 1) 2 ( y 2) 2 ( z 3) 2 362. center (6, 0, 5), radius 13. center (9, 7, 4), radius 154. center (8, 1, 1), radius 2215Chapter 15 ● Skills Practice 969

5. center (4, 0, 0), radius 86. center (5, 3, 6), radius 12Determine the center and radius of each sphere given the equation of the sphere.7. (x 11) 2 ( y 2) 2 z 2 9center (11, 2, 0), radius 38. (x 9) 2 ( y 11) 2 (z 3) 2 369. (x 2) 2 ( y 17) 2 (z 8) 2 19610. (x 10) 2 y 2 (z 19) 2 8111. x 2 y 2 (z 2) 2 2512. (x 14) 2 ( y 20) 2 (z 7) 2 625Write each equation of a sphere in standard form.13. x 2 y 2 z 2 2x 10y 4z 6 0( x 2 2 x 1) ( y 2 10y 25) ( z 2 4z 4) 6 1 25 4( x 1) 2 ( y 5) 2 ( z 2) 2 3614. x 2 y 2 z 2 4x 20z 4 0© 2010 Carnegie Learning, Inc.1515. x 2 y 2 z 2 8x 12y 6z 12 0970 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________16. 2x 2 2y 2 2z 2 24x 40y 16 017. 4x 2 4y 2 4z 2 64x 88y 8z 488 018. 3x 2 3y 2 3z 2 90y 24z 423 0© 2010 Carnegie Learning, Inc.Determine the intersection of each given sphere with the given plane.19. x 2 y 2 z 2 36 and y 0x 2 y 2 z 2 36x 2 0 2 z 2 36x 2 z 2 36The intersection is the great circle x 2 z 2 36.15Chapter 15 l Skills Practice 971

20. x 2 y 2 z 2 36 and z 821. (x 1) 2 ( y 4) 2 z 2 100 and x 722. (x 8) 2 ( y 2) 2 (z 3) 2 625 and y 223. (x 3) 2 ( y 6) 2 (z 4) 2 289 and z 4© 2010 Carnegie Learning, Inc.15972 Chapter 15 l Skills Practice

Name _____________________________________________Date ____________________24. (x 7) 2 ( y 10) 2 (z 5) 2 25 and y 5Determine whether the intersection of each sphere and plane is a point,a circle, or a great circle.25. x 2 y 2 z 2 2500 and y 40x 2 y 2 z 2 2500x 2 40 2 z 2 2500x 2 1600 z 2 2500x 2 z 2 900The intersection is a circle.26. (x 6) 2 ( y 8) 2 (z 1) 2 169 and z 1© 2010 Carnegie Learning, Inc.27. x 2 ( y 5) 2 (z 3) 2 2500 and y 4515Chapter 15 l Skills Practice 973

28. (x 1) 2 ( y 1) 2 z 2 289 and x 729. (x 4) 2 y 2 (z 2) 2 1681 and z 230. (x 2) 2 ( y 3) 2 (z 4) 2 100 and x 8© 2010 Carnegie Learning, Inc.15974 Chapter 15 l Skills Practice