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All Textbook Solutions for Precalculus

For Exercises 9-10, a. Use the distance formula to find the distances d1,d2,d3 and d4 . b. Find the sum d1+d2 . c. Find the sum d3+d4 . d. How do the sums from parts (b) and (c) compare? e. How do the sums of the distances from parts (b) and (c) relate to the length of the major axis?For Exercises 11-12, from the equation of the ellipse, determine if the major axis is horizontal or vertical. a. x22+y25=1 b. x25+y22=1 For Exercises 11-12, from the equation of the ellipse, determine if the major axis is horizontal or vertical. a. x211+y210=1 b. x210+y211=1 For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) x2100+y225=1 For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) x264+y249=1 For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) x225+y2100=1 For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) x249+y264=1 For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) 4x2+25y2=100 For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) 9x2+64y2=576 19PE For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) 64x216y2=64 21PE For Exercises 13-22, a. Identify the center of the ellipse. b. Determine the value of a . c. Determine the value of b . d. Identify the vertices. e. Identify the endpoints of the minor axis. f. Identify the foci. g. Determine the length of the major axis. h. Determine the length of the minor axis. i. Graph the ellipse. (See Examples 1-2) x218+y27=1 23PE 24PE For Exercises 23-32, a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse. (See Example 3) x+4249+y2264=1 For Exercises 23-32, a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse. (See Example 3) x+1236+y5281=1 27PE For Exercises 23-32, a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse. (See Example 3) x22+y24=1 29PE For Exercises 23-32, a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse. (See Example 3) x2+4y+42=100 For Exercises 23-32, a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse. (See Example 3) 4x3225+16y2249=1 For Exercises 23-32, a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse. (See Example 3) 4x4281+16y32225=1 For Exercises 33-42, a. Write the equation of the ellipse in standard form. (See Example 4) b. Identify the center, vertices, endpoints of the minor axis, and foci. 3x2+5y2+12x60y+177=0 For Exercises 33-42, a. Write the equation of the ellipse in standard form. (See Example 4) b. Identify the center, vertices, endpoints of the minor axis, and foci. 7x2+11y2+70x66y+197=0 For Exercises 33-42, a. Write the equation of the ellipse in standard form. (See Example 4) b. Identify the center, vertices, endpoints of the minor axis, and foci. 3x2+2y230x4y+59=0 36PE 37PE 38PE For Exercises 33-42, a. Write the equation of the ellipse in standard form. (See Example 4) b. Identify the center, vertices, endpoints of the minor axis, and foci. 36x2+100y2180x+800y+925=0 40PE 41PE For Exercises 33-42, a. Write the equation of the ellipse in standard form. (See Example 4) b. Identify the center, vertices, endpoints of the minor axis, and foci. 25x2+16y2+64y+63=0 43PE 44PE For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Endpoints of minor axis:17,0 and 17,0 Foci:0,9 and 0,9 46PE For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Major axis parallel to the x-axis;Center:2,3 ; Length of major axis:14 units; Length of minor axis:10 units 48PE For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Foci:0,1 and8,1 ; Length of minor axis; 6 units 50PE For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Vertices:0,5 and 0,5 ; Passes through 165,3 52PE 53PE For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Vertices:4,5 and 4,5 ; Foci:6,5 and 6,5 For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Vertices:2,1 and 12,1 ; Foci:533,1 and 5+33,1 For Exercises 43-56, write the standard form of an equation of an ellipse subject to the given conditions. Vertices:8,4 and 2,4 ; Foci:321,4 and 3+21,4 The reflective property of an ellipse is used in lithotripsy. Lithotripsy is a technique for treating kidney stones without surgery. Instead, high-energy shock waves are emitted from one focus of an elliptical shell and reflected painlessly to a patient's kidney stone located at the other focus. The vibration from the shock waves shatter the stone into pieces small enough to pass through the patient's urine. A vertical cross section of a lithotripter is in the shape of a semiellipse with the dimensions shown. Approximate the distance from the center along the major axis where the patient's kidney stone should be located so that the shock waves will target the stone. Round to 2 decimal places. (See Example 6)The reflective property of an ellipse is the principle behind "whispering galleries." These are rooms with elliptically shaped ceilings such that a person standing at one focus can hear even the slightest whisper spoken by another person standing at the other focus. Suppose that a dome has a semielliptical ceiling, 96ft long and 23ft high. Choose a coordinate system so that the center of the semiellipse is 0,0 with vertices 48,0 and 48,0 and with the top of the ceiling at 0,23 . Approximately how far from the center along the major axis should each person be standing to hear the "whispering" effect? Round to 1 decimal place.A homeowner wants to make an elliptical rug from a 12-ft by 10-ft rectangular piece of carpeting. a. What lengths of the major and minor axes would maximize the area of the new rug? b. Write an equation of the ellipse with maximum area. Use a coordinate system with the origin at the center of the rug and horizontal major axis. c. To cut the rectangular piece of carpeting, the homeowner needs to know the location of the foci. Then she will insert tacks at the foci, take a piece of string the length of the major axis, and fasten the ends to the tacks. Drawing the string tight, she'll use a piece of chalk to trace the ellipse. At what coordinates should the tacks be located? Describe the location.60PE Charles and Bernice (â€œRay") Eames were American designers who made major contributions to modern architecture and furniture design. Suppose that a manufacturer wants to make an Eames elliptical coffee table 90 in. long and 30in. wide out of an 8-ft by 4-ft piece of birch plywood. If the center of a piece of plywood is positioned 0,0 , determine the distance from the center at which the foci should be located to draw the ellipse.A window above a door is to be made in the shape of a semiellipse. If the window is 10ft at the base and 3ft high at the center, determine the distance from the center at which the foci are located.63PE Choose one: An ellipse with eccentricity close to 0 will appear a. more elongated. b. more circular.Choose one: An ellipse with eccentricity close to 1 will appear a. more elongated. b. more circular.Choose one: If the foci of an ellipse are close to the center of the ellipse, then the graph will appear a. more elongated. b. more circular.For Exercises 67-72, determine the eccentricity of the ellipse. x2169+y225=1 For Exercises 67-72, determine the eccentricity of the ellipse. x2100+y264=1 69PE For Exercises 67-72, determine the eccentricity of the ellipse. x1329+y+79225=1 For Exercises 67-72, determine the eccentricity of the ellipse. x212+y326=1 For Exercises 67-72, determine the eccentricity of the ellipse. x+7218+y212=1 Halley's Comet and the Earth both orbit the Sun in elliptical paths with the Sun at one focus. The eccentricity of the comets orbit is 0.967 and the eccentricity of the Earth's orbit is0.0167 .The eccentricity for the Earth is close to zero, whereas the eccentricity for Halley's Comet is close to 1 . Based on this information, how do the orbits compare?Halley's Comet and the comet Hale-Bopp both orbit the Sun in elliptical paths with the Sun at one focus. The eccentricity of Halley's Comet's orbit is 0.967 and the eccentricity of comet Hale-Bopp s orbit is 0.995 . Which comet has a more elongated orbit?The moon's orbit around the Earth is elliptical with the Earth at one focus and with eccentricity0.0549 . If the distance between the moon and Earth at the closest point is 363,300 km, determine the distance at the farthest point. Round to the nearest 100 km. (See Example 7)The planet Saturn orbits the Sun in an elliptical path with the Sun at one focus. The eccentricity of the orbit is 0.0565 and the distance between the Sun and Saturn at perihelion (the closest point) is 1.353109km . Determine the distance at aphelion (the farthest point). Round to the nearest million kilometers.The Roman Coliseum is an elliptical stone and concrete amphitheater in the center of Rome, built between 70A.D. and 80A.D . The Coliseum seated approximately 50,000 spectators and was used among other things for gladiatorial contests. a. Using a vertical major axis, write an equation of the ellipse representing the center arena if the maximum length is 287ft and the maximum width is 180ft . Place the origin at the center of the arena. b. Approximate the eccentricity of the center arena. Round to 2 decimal places. c. Find an equation of the outer ellipse if the maximum length is 615ft and the maximum width is 510ft d. Approximate the eccentricity of the outer ellipse. Round to 2 decimal places. e. Explain how you know that the outer ellipse is more circular than the inner ellipse.The Ellipse, also called Presidents Park South, is a park in Washington, D.C. The lawn area is elliptical with a major axis of 1058ft and minor axis of 903ft . a. Find an equation of the elliptical boundary. Take the horizontal axes to be the major axis and locate the origin of the coordinate system at the center of the ellipse. b. Approximate the eccentricity of the ellipse. Round to 2 decimal places.79PE For Exercises 79-82, write the standard form of an equation of the ellipse subject to the following conditions. Center: 0,0; Eccentricity: 4041; Major axis vertical of length 82 units For Exercises 79-82, write the standard form of an equation of the ellipse subject to the following conditions. Foci: 0,1 and 8,1; Eccentricity: 45 82PE A circular vent pipe is placed on a flat roof. a. Write an equation of the circular cross section that the pipe makes with the roof. Assume that the origin is placed at the center of the circle. b. Now suppose that the pipe is placed on a roof with a slope of 35. What shape will the cross section of the pipe form with the plane of the roof? c. Determine the length of the major and minor axes. Find the exact value and approximate to 1 decimal place if necessary.A cylindrical glass of water with diameter 3.5in. sits on a horizontal counter top. a. Write an equation of the circular surface of the water. Assume that the origin is placed at the center of the circle. b. If the glass is tipped 30, what shape will the surface of the water have? c. With the glass tipped 30, the waterline makes a slope of 12 with the coordinate system shown. Determine the length of the major and minor axes. Round to 1 decimal place.85PE The graph of x216+y281=1 represents an ellipse. Determine the part of the ellipse represented by the given equation. a. x=41y281 b. x=41y281 c. y=91x216 d. y=91x216 For Exercises 87-90, solve the system of equations. x225+y29=13x+5y=15 88PE For Exercises 87-90, solve the system of equations. x24+y216=1y=x2+4 For Exercises 87-90, solve the system of equations. x264+y2289=1y=1364x213 Given an ellipse with major axis of length 2a and minor axis of length 2b, the area is given by A=ab. The perimeter is approximated by p2a2+b2. for Exercises 91-92, a. Determine the area of the ellipse. b. Approximate the perimeter. x28+y+324=1 92PE A line segment with endpoints on an ellipse, perpendicular to the major axis, and passing through a focus, is called a lotus rectum of the ellipse. Show that the length of a latus rectum is 2b2a for the ellipse. x2a2+y2b2=1 [Hint Substitute c,y into the equation and solve for y. Recall that c2=a2+b2. ]94PE 95PE Given an equation of an ellipse in standard form, how do you determine whether the major axis is horizontal or vertical?Explain the difference between the graphs of the two equations. 4x2+9y2=36 , 4x+9y=36 Discuss the solution set of the equation. x429+x+2216=1 This exercise guides you through the steps to find the standard form of an equation of an ellipse centered at the origin with foci on the x-axis. a. Refer to the figure to verify that the distance from F1toV2isa+c and the distance from F2toV2isac. is Verify that the sum of these distances is 2a. b. Write an expression that represents the sum of the distances fromF1 to x,y and from F2 tox,y. Then set this expression equal to 2a. c. Given the equation x+c2+y2+xc2+y2=2a, isolate the leftmost radical and square both sides of the equation. Show that the equation can be written asaxc2+y2=a2xc. d. Square both sides of the equation axc2+y2=a2xc and show that the equation can be written as a2c2x2+a2y2=a2a2c2. (Hint. Collect variable terms on the left side of the equation and constant terms on the right side.) e. Replace a2c2 by b2. Then divide both sides of the equation by a2b2. Verify that the resulting equation is x2a2+y2b2=1.Find the points on the ellipse that are twice the distance from one focus to the other. x225+y29=1 101PE 102PE For Exercises 102-105, graph the ellipse from the given exercise on a square viewing window. Exercise 15 For Exercises 102-105, graph the ellipse from the given exercise on a square viewing window. Exercise 36 For Exercises 102-105, graph the ellipse from the given exercise on a square viewing window. Exercise 35 Repeat Example 1 with the equation: x29y24=1 2SP Repeat Example 3 with the equation: x+124+y4225=1 4SP 5SP Determine the eccentricity of the hyperbola. a.y225x2144=1b.y2144x225=1 7SP A is the set of points x,y in a plane such that the difference in distances between x,y and two fixed points (called ) is a positive constant.The points where a hyperbola intersects the line through the foci are called the .3PE 4PE The equation x2a2y2b2=1 represents a hyperbola with a (horizontal/vertical) transverse axis. The vertices are given by the ordered pairs and . The asymptotes are given by the equations and .The equation y2a2x2b2=1 represents a hyperbola with a (horizontal/vertical) transverse axis. The vertices are given by the ordered pairs and . The asymptotes are given by the equations and .For Exercises 7-8, a. Use the distance formula to find the distances d1,d2,d3 and d4 . b. Find the difference of the distances: d1d2 . c. Find the difference of the distances: d3d4 . d. How do the results from parts (b) and (c) compare? e. How do the results of part (b) and part (c) compare to the length of the transverse axis?For Exercises 7-8, a. Use the distance formula to find the distances d1,d2,d3 and d4 . b. Find the difference of the distances: d1d2 . c. Find the difference of the distances: d3d4 . d. How do the results from parts (b) and (c) compare? e. How do the results of part (b) and part (c) compare to the length of the transverse axis?For Exercises 9-12, determine whether the transverse axis and foci of the hyperbola are on the x-axis or the y-axis . x215y220=1 For Exercises 9-12, determine whether the transverse axis and foci of the hyperbola are on the x-axis or the y-axis . y212x218=1 For Exercises 9-12, determine whether the transverse axis and foci of the hyperbola are on the x-axis or the y-axis . x2+y23=1 12PE 13PE For Exercises 13-22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) x225y236=1 For Exercises 13-22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) y24x236=1 For Exercises 13-22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) y29x249=1 17PE For Exercises 13-22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) 49y216x2=784 19PE For Exercises 13-22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) 7x211y2=77 21PE For Exercises 13-22, a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Examples 1-2) 4x28116y2225=1 For Exercises 23-32, a. Identify1he center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Example 3) x429y+2216=1 For Exercises 23-32, a. Identify1he center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Example 3) x3236y+1264=1 25PE For Exercises 23-32, a. Identify1he center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Example 3) y4236x+5216=1 For Exercises 23-32, a. Identify1he center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Example 3) 100y7281x+42=8100 28PE For Exercises 23-32, a. Identify1he center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Example 3) y2x3212=1 30PE 31PE For Exercises 23-32, a. Identify1he center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola. (See Example 3) x2y6224=1 For Exercises 33-40, a. Write the equation of the hyperbola in standard form. (See Example 4) b. Identify the center, vertices, and foci. 7x25y2+42x+10y+23=0 For Exercises 33-40, a. Write the equation of the hyperbola in standard form. (See Example 4) b. Identify the center, vertices, and foci. 5x23y2+10x+24y73=0 For Exercises 33-40, a. Write the equation of the hyperbola in standard form. (See Example 4) b. Identify the center, vertices, and foci. 5x2+9y2+20x72y+79=0 For Exercises 33-40, a. Write the equation of the hyperbola in standard form. (See Example 4) b. Identify the center, vertices, and foci. 7x2+16y270x+96y143=0 37PE 38PE For Exercises 33-40, a. Write the equation of the hyperbola in standard form. (See Example 4) b. Identify the center, vertices, and foci. 36x2+64y2+108x+256y401=0 For Exercises 33-40, a. Write the equation of the hyperbola in standard form. (See Example 4) b. Identify the center, vertices, and foci. 144x2+25y2+720x50y4475=0 For Exercises 41-50, write the standard form of the equation of the hyperbola subject to the given conditions. (See Example 5) Vertices: 12,0 , 12,0 ; Foci: 13,0,13,0 For Exercises 41-50, write the standard form of the equation of the hyperbola subject to the given conditions. (See Example 5) Vertices: 40,0,40,0 ; Foci: 41,0,41,0 43PE 44PE 45PE 46PE 47PE 48PE For Exercises 41-50, write the standard form of the equation of the hyperbola subject to the given conditions. (See Example 5) Corners of the reference rectangle: 8,7,6,7,8,3,63 ; Horizontal transverse axis 50PE For Exercises 51-52, a. Determine the eccentricity of each hyperbola. b. Based on the eccentricity, match the equation with its graph. The scaling is the same for both graphs. Equation 1: x2144y281=1 Equation 2: x281y2144=1 For Exercises 51-52, a. Determine the eccentricity of each hyperbola. (See Example 6) b. Based on the eccentricity, match the equation with its graph. The scaling is the same for both graphs. Equation 1: x2225y264=1 Equation 2: x264y2225=1 53PE For Exercises 53-54, determine the eccentricity. y3.8249x2.72576=1 Determine the eccentricity of a hyperbola with a horizontal transverse axis of length 24 units and asymptotes y=34x.56PE Determine the standard form of an equation of a hyperbola with eccentricity 54 and vertices1,1 and 7,1 .Determine the standard form of an equation of a hyperbola with eccentricity 1312 and vertices2,8 and 2,16 .Two radio transmitters are 1000 mi apart at points A and B along a coastline. Using LORAN on the ship, the time difference between the radio signals is 4 milliseconds 0.004sec .If radio signals travel 186mi/millisecond, find an equation of the hyperbola with foci at A and B, on which the ship is located.Suppose that two microphones 1500m apart at points A and B detect the sound of a rifle shot The time difference between the sound detected at point A and the sound detected at point B is 4 sec. If sound travels at approximately 330m/sec , find an equation of the hyperbola with foci at A and B defining the points where the shooter may be located.In some designs of eyeglasses, the surface is "aspheric," meaning that the contour varies slightly from spherical. An aspheric lens is often used to correct for spherical aberrationâ€” a distortion due to increased refraction of light rays when they strike the lens near its edge. Aspheric lenses are often designed with hyperbolic cross sections. Write an equation of the cross section of the hyperbolic lens shown if the center is 0,0 one vertex is2,0 ,and the focal length (distance between center and foci) is 85 . Assume that all units are in millimeters.In 1911,Ernest Rutherford discovered the nucleus of the atom. Experiments leading to this discovery involved the scattering of alpha particles by the heavy nuclei in gold foil. When alpha particles are thrust toward the gold nuclei, the particles are deflected and follow a hyperbolic path. Suppose that the minimum distance that the alpha particles get to the gold nuclei is 4 microns (1 micron is one-millionth of a meter) and that the hyperbolic path has asymptotes of y=12x Determine an equation of the path of the particles shown.Atomic particles with like charges tend to repel one another. Suppose that two beams of like-charged particles are hurled toward each other from two parallel atomic accelerators. The path defined by the particles is x24y2=36 , where x and y are measured in microns. What is the minimum distance between the particles?A returning boomerang is a V-shaped throwing device made from two wings that are set at a slight tilt and that have an airfoil design. One side is rounded and the other side is flat,similar to an airplane propeller. When thrown properly, a boomerang follows a circular flight path and should theoretically return close to the point of release. The boomerang pictured is approximately in the shape of one branch of a hyperbola (although the two wings are in slightly different planes). To construct the hyperbola, an engineer needs to know the location of the foci. Determine the location of the focus to the right of the center if the vertex is7.5in . from the center and the equations of the asymptotes are y=45x Round the coordinates to the nearest tenth of an inch.In September 2009, Australian astronomer Robert H.McNaught discovered comet C/2009 R1 (McNaught). The orbit of this comet is hyperbolic with the Sun at one focus. Because the orbit is not elliptical, the comet will not be captured by the Sun's gravitational pull and instead will pass by the Sun only once. The comet reached perihelion on July 2, 2010 The path of comet can be modeled by the equation x21191.22y230.92=1 where x and y are measured in AU (astronomical units). a. Determine the distance (in AU) at perihelion. Round to 1 decimal place. b. Using the rounded value from part (a), if 1 AUâ‰ˆ93,000,000 mi, find the distance in miles.The cross section of a cooling tower of a nuclear power plant is in the shape of a hyperbola, and can be modeled by the equation. x2625y8022500=1 where x and y are measured in meters. The top of the tower is 120m above the base. a. Determine the diameter of the tower at the base. Round to the nearest meter. b. Determine the diameter of the tower at the top. Round to the nearest meter.67PE 68PE 69PE 70PE For Exercises 71-74, find the standard form of the equation of the ellipse or hyperbola shown.72PE 73PE For Exercises 71-74, find the standard form of the equation of the ellipse or hyperbola shown.For Exercises 75-76, solve the system of equations. 9x24y2=3613x2+8y2=8 For Exercises 75-76, solve the system of equations. x2+4y2=364x2y2=8 Given an equation of a hyperbola in standard form, how do you determine whether the transverse axis is horizontal or vertical?78PE 79PE 80PE A line segment with endpoints on a hyperbola, perpendicular to the transverse axis, and passing through a focus is called a latus rectum of the hyperbola (shown in red). Show that the length of a latus rectum is 2b2a for the hyperbola x2a2y2b2=1 82PE 83PE a. Radio signals emitted from points 8,0 and 8,0 indicate that a plane is 8mi closer to 8,0 than to (8,0) . Find an equation of the hyperbola that passes through the plane's location and with foci 8,0 and 8,0 . All units are in miles. b. At the same time, radio signals emitted from points 0,8 and 0,8 indicate that the plane is 4mi farther from 0,8 than from 0,8 . Find an equation of the hyperbola that passes through the plane's location and with foci 0,8 and 0,8 . c. From the figure, the plane is located in the fourth quadrant of the coordinate system. Solve the system of equations defining the two hyperbolas for the point of intersection in the fourth quadrant. This is the location of the plane. Then round the coordinates to the nearest tenth of a mile.85PE 86PE 87PE For Exercises 85-88, graph the hyperbola from the given exercise. Exercise 38 Given a parabola defined by x2=4y , find the focus and an equation of the directrix.Repeat Example 2 with the equation: x2=8y 3SP 4SP Repeat Example 5 with the equation: 4x212x12y+21=0 6SP Repeat Example 7 with a radio telescope of diameter 70m and depth 14m .A is the set of all points in a plane that are equidistant from a fixed line (called the ) and a fixed point (called the ).2PE The of a parabola is the point of intersection of the parabola and the axis of symmetry.The distance between the vertex and the focus of a parabola is called the length and is often represented by p .Given y2=4px with p0, the parabola opens (upward/downward/left/right).Given x2=4py with p0, the parabola opens (upward/downward/left/right).The line segment perpendicular to the axis of symmetry, passing through the focus and with endpoints on the parabola is called the .The length of the latus rectum is called the diameter.Given xh2=4pyk , the ordered pairs representing the vertex and focus are and respectively. The directrix is the line defined by the equation .Given yk2=4pxh, the ordered pairs representing the vertex and focus are and respectively. The directrix is the line defined by the equation .If the directrix is horizontal, then the parabola opens (horizontally/vertically).If the line of symmetry is horizontal, then the parabola opens (horizontally/vertically).13PE For Exercises 13-14, the graph of a parabola is given. a. Determine the distances d1,d2,d3 and d4. b. Compare d1, and d2. c. Compare d3 and d4.For Exercises 15-22, a model of the form x2=4py or y2=4px is given. a. Determine the value of p. b. Identify the focus of the parabola. c. Write an equation for the directrix. (See Example 1) x2=24y For Exercises 15-22, a model of the form x2=4py or y2=4px is given. a. Determine the value of p . b. Identify the focus of the parabola. c. Write an equation for the directrix. (See Example 1) x2=12y 17PE 18PE 19PE For Exercises 15-22, a model of the form x2=4py or y2=4px is given. a. Determine the value of p. b. Identify the focus of the parabola. c. Write an equation for the directrix. (See Example 1) x2=11y For Exercises 15-22, a model of the form x2=4py or y2=4px is given. a. Determine the value of p. b. Identify the focus of the parabola. c. Write an equation for the directrix. (See Example 1) x=y2 22PE A 20-in. satellite dish for a television has parabolic cross sections. A coordinate system is chosen so that the vertex of a cross section through the center of the dish is located at 0,0. The equation of the parabola is modeled by x2=25.2y, where x and y are measured in inches a. Where should the receiver be placed to maximize signal strength? That is, where is the focus? (See Example 1) b. Determine the equation of the directrix.Solar cookers provide an alternative form of cooking in regions of the world where consistent sources of fuel are not readily available. Suppose that a 36-in. solar cooker has parabolic cross sections. A coordinate system is chosen with the origin placed at the vertex of a cross section through the center of the mirror. The equation of the parabola is modeled by x2=82y, where x and y are measured in inches a. Where should a pot be placed to maximize heat? That is, where is the focus? b. Determine the equation of the directrix.If a cross section of the parabolic mirror in a flashlight has an equation y2=2x, where should the bulb be placed?A cross section of the parabolic mirror in a car headlight is modeled by y2=12x. Where should the bulb be placed?For Exercises 27-34, an equation of a parabola x2=4py or y2=4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum, c. Graph the parabola. d. Write equations for the directrix and axis of symmetry x2=4y For Exercises 27-34, an equation of a parabola x2=4py or y2=4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum, c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 2-3) x2=20y For Exercises 27-34, an equation of a parabola x2=4py or y2=4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 2-3) 10y2=80x For Exercises 27-34, an equation of a parabola x2=4py or y2=4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 2-3) 3y2=12x For Exercises 27-34, an equation of a parabola x2=4py or y2=4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum, c. Graph the parabola. d. Write equations for the directrix and axis of symmetry (See Example 2-3) 4x2=40y 32PE 33PE For Exercises 27-34, an equation of a parabola x2=4py or y2=4px is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum, c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 2-3) y2=2x 35PE For Exercises 35-44, an equation of a parabola xh2=4pykoryk2=4pxh is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) y+42=16x2 37PE For Exercises 35-44, an equation of a parabola xh2=4pykoryk2=4pxh is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) x52=8y+2 39PE For Exercises 35-44, an equation of a parabola xh2=4pykoryk2=4pxh is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) x+22=y74 For Exercises 35-44, an equation of a parabola xh2=4pykoryk2=4pxh is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) y32=7x14 For Exercises 35-44, an equation of a parabola xh2=4pykoryk2=4pxh is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) y62=9x34 For Exercises 35-44, an equation of a parabola xh2=4pykoryk2=4pxh is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. (See Example 4) 2y3=110x+62 44PE 45PE For Exercises 45-52, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identify the vertex, focus, and focal diameter. (See Example 5) x24x8y20=0 47PE 48PE For Exercises 45-52, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identify the vertex, focus, and focal diameter. (See Example 5) 4x228x+24y+73=0 For Exercises 45-52, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identify the vertex, focus, and focal diameter. (See Example 5) 4x2+36x+40y+1=0 For Exercises 45-52, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identify the vertex, focus, and focal diameter. (See Example 5) 16y2+24y16x+57=0 For Exercises 45-52, an equation of a parabola is given. a. Write the equation of the parabola in standard form. b. Identify the vertex, focus, and focal diameter. (See Example 5) 16y256y16x+81=0 53PE 54PE For Exercises 53-58, fill in the blanks. Let p represent the focal length (distance between the vertex and focus). If the vertex of a parabola is 3,0 and the focus is 7,0, then the directrix is given by the equation , and the value of p is .For Exercises 53-58, fill in the blanks. Let p represent the focal length (distance between the vertex and focus). If the vertex of a parabola is 4,2 and the focus is 4,7, then the directrix is given by the equation , and the value of p is .For Exercises 53-58, fill in the blanks. Let p represent the focal length (distance between the vertex and focus). If the focal length of a parabola is 3 units, then the length of the latus rectum is units.58PE If the focal length of a parabola is 6 units and the vertex is 3,2, is it possible to determine whether the parabola opens upward, downward, to the right, or to the left? Explain.60PE 61PE 62PE For Exercises 61-68, determine the standard form of an equation of the parabola subject to the given conditions (See Example 6) Focus: 2,4; Vertex: 2,1 64PE For Exercises 61-68, determine the standard form of an equation of the parabola subject to the given conditions (See Example 6) Focus: 6,2; Directrix: y=0 For Exercises 61-68, determine the standard form of an equation of the parabola subject to the given conditions (See Example 6) Focus: 4,5; Directrix: x=0 67PE For Exercises 61-68, determine the standard form of an equation of the parabola subject to the given conditions (See Example 6) Vertex: 4,2; Parabola passes through 8,14 Suppose that a solar cooker has a parabolic mirror (see figure). a. Use a coordinate system with origin at the vertex of the mirror and write an equation of a parabolic cross section of the mirror. b. Where should a pot be placed so that it receives maximum heat? (See Example 7)A solar water heater is made from a long sheet of metal bent so that the cross sections are parabolic. A long tube of water is placed inside the curved surface so that the height of the tube is equal to the focal length of the parabolic cross section. In this way, water in the tube is exposed to maximum heat. a. Determine the focal length of the parabolic cross sections so that the engineer knows where to place the tube. b. Use a coordinate system with origin at the vertex of a parabolic cross section and write an equation of the parabola.The Hubble Space Telescope was launched into space in 1990 and now orbits the Earth at 5mi/sec at a distance of 353mi above the Earth. From its location in space, the Hubble is free from the distortion of the Earth's atmosphere, enabling it to return magnificent images from distant stars and galaxies. The quality of the Hubble's images results from a large parabolic mirror, 2.4mi 7.9ft in diameter, that collects light from space. Suppose that a coordinate system is chosen so that the vertex of a cross section through the center of the mirror is located at 0,0 . Furthermore, the focal length is 57.6m . a. Assume that x and y are measured in meters. Write an equation of the parabolic cross section of the mirror for 1.2x1.2 See figure. b. How thick is the mirror at the edge? That is, what is the y value for x=1.2 ?The James Webb Space Telescope (JWST) is a new space telescope currently under construction. The JWST will orbit the Sun in an orbit roughly 1.5 million km from the earth, in a position with the Earth between the telescope and the Sun. The primary mirror of the JWST will consist of 18 hexagonally shaped segments that when fitted together will be 6.5 min diameter (over 2.5 times the diameter of the Hubble). With a larger mirror, the JWST will be able to collect more light to â€œsee" deeper into space. The primary mirror when pieced together will function as a parabolic mirror with a focal length of 131.4m . Suppose that a coordinate system is chosen with 0,0 at the center of a cross section of the primary mirror. Write an equation of the parabolic cross section of the mirror for 3.25x3.25 . Assume that x and y are measured in meters.The Subaru telescope is a large optical-infrared telescope at the summit of Mauna Kea, Hawaii. The telescope has a parabolic mirror 8.2m in diameter with a focal length of 15m . a. Suppose that a cross section of the mirror is taken through the vertex, and that a coordinate system is set up with 0,0 placed at the vertex. If the focus is 0,15, find an equation representing the curve. b. Determine the vertical displacement of the mirror relative to horizontal at the edge of the mirror. That is, find the y value at a point 4.1m to the left or right of the vertex. c. What is the average slope between the vertex of the parabola and the point on the curve at the right edge?A parabolic mirror on a telescope has a focal length of 16cm . a. For the coordinate system shown, write an equation of the parabolic cross section of the mirror. b. Determine the displacement of the mirror relative to the y-axis at the edge of the mirror. That is, find the x value at a point 12cm above or below the vertex.75PE A cable hanging freely between two vertical support beams forms a curve called a catenary. The shape of a catenary resembles a parabola but mathematically the two functions are quite different. a. On a graphing utility, graph a catenary defined by y=12ex+ex and graph the parabola defined by y=x2+1 . b. A catenary and a parabola are so similar in shape that we can often use a parabolic curve to approximate the shape of a catenary. For example, a bridge has cables suspended from a larger approximately parabolic cable. Take the origin at a point on the road directly below the vertex and write an equation of the parabolic cable. c. Determine the focal length of the parabolic cable. d. Determine the length of the vertical support cable 100ft from the vertex. Round to the nearest tenth of a foot.77PE For Exercises 77-78, solve the system of nonlinear equations. y22=8x12xy=8 Given an equation of a parabola xh2=4pykoryk2=4pxh, how can you determine whether the parabola opens vertically or horizontally?80PE The surface defined by the equation z=4x2+y2 is called an elliptical paraboloid. a. Write the equation with x=0. What type of curve is represented by this equation? b. Write the equation with y=0. What type of curve is represented by this equation? c. Write the equation with z=0. What type of curve is represented by this equation?A jet flies in a parabolic arc to simulate partial weightlessness. The curve shown in the figure represents the plane's height y (in 1000ft ) versus the time t (in sec). a. For each ordered pair, substitute the t and y values into the model y=at2+bt+c to form a linear equation with three unknowns a,b,andc .Together, these form a system of three linear equations with three unknowns. b. Use a graphing utility to solve for a,b,andc. c. Substitute the known values of a,b,andc into the model y=at2+bt+c . d. Determine the vertex of the parabola. e. Determine the focal length of the parabola.For Exercises 83-86, graph the parabola from the given exercise. Exercise 29 For Exercises 83-86, graph the parabola from the given exercise. Exercise 30 For Exercises 83-86, graph the parabola from the given exercise. Exercise 41 For Exercises 83-86, graph the parabola from the given exercise. Exercise 42 For Exercises 1-8, identify each equation as representing a circle, an ellipse, a hyperbola, or a parabola. If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, equation of the directrix, and equation of the axis of symmetry. x2216y+229=1 2PRE 3PRE For Exercises 1-8, identify each equation as representing a circle, an ellipse, a hyperbola, or a parabola. If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, equation of the directrix, and equation of the axis of symmetry. x32+y+72=25 For Exercises 1-8, identify each equation as representing a circle, an ellipse, a hyperbola, or a parabola. If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, equation of the directrix, and equation of the axis of symmetry. 16x+12+y2=16 For Exercises 1-8, identify each equation as representing a circle, an ellipse, a hyperbola, or a parabola. If the equation represents a circle, identify the center and radius. If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity. If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity. If the equation represents a parabola, identify the vertex, focus, equation of the directrix, and equation of the axis of symmetry. x12=10y+3 7PRE 8PRE For Exercises 9-16, determine whether the equation represents a circle, an ellipse, a hyperbola, or a parabola. Write the equation in standard form. 9x216y236x64y172=0 For Exercises 9-16, determine whether the equation represents a circle, an ellipse, a hyperbola, or a parabola. Write the equation in standard form. 9x2+25y2100y125=0 11PRE For Exercises 9-16, determine whether the equation represents a circle, an ellipse, a hyperbola, or a parabola. Write the equation in standard form. x2+y26x+14y+33=0 13PRE 14PRE 15PRE For Exercises 9-16, determine whether the equation represents a circle, an ellipse, a hyperbola, or a parabola. Write the equation in standard form. 144x2+25y21152x50y5879=0

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