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All Textbook Solutions for Precalculus: Mathematics for Calculus - 6th Edition
32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52EParabolic Reflector A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is 12 cm. (a) Find an equation of the parabola. (b) Find the diameter d(C, D) of the opening, 20 cm from the vertex.Satellite Dish A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus F. The reflector is 1 ft deep and 20 ft wide from rim to rim (see the figure). How far is the receiver from the vertex of the parabolic reflector?Suspension Bridge In a suspension bridge the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 600 m apart, and the lowest point of the suspension cables is 150 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex. [Note: This equation is used to find the length of cable needed in the construction of the bridge.]Reflecting Telescope The Hale telescope at the Mount Palomar Observatory has a 200-in. mirror, as shown in the figure. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.57E58EAn ellipse is the set of all points in the plane for which the __________ of the distances from two fixed points F1 and F2 is constant. The points F1, and F2 are called the __________ of the ellipse.The graph of the equation x2a2+y2b2=1 with a b 0 is an ellipse with vertices (_____, _____ ) and (_____, _____) and foci ( c, 0), where c = __________. So the graph of x252+y242=1 is an ellipse with vertices (_____, _____ ) and (_____, _____ ) and foci (_____, _____) and (_____, _____).The graph of the equation x2b2+y2a2=1 with a b 0 is an ellipse with vertices (_____, _____ ) and (_____, _____) and foci (0, c), where c = __________. So the graph of x242+y252=1is an ellipse with vertices (_____, _____) and (_____, _____) and foci (_____, _____ ) and (_____, _____).Label the vertices and foci on the graphs given for the ellipses in Exercises 2 and 3. (a) x252+y242=1 (b) x242+y252=15E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50EPerihelion and Aphelion The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earths distance from the sun is 147,000,000 km at perihelion and 153,000,000 km at aphelion. Find an equation for the earths orbit. (Place the origin at the center of the orbit with the sun on the x-axis.)52ELunar Orbit For an object in an elliptical orbit around the moon, the points in the orbit that are closest to and farthest from the center of the moon arc called perilune and apolune, respectively. These are the vertices of the orbit. The center of the moon is at one focus of the orbit. The Apollo 11 spacecraft was placed in a lunar orbit with perilune at 68 mi and apolune at 195 mi above the surface of the moon. Assuming that the moon is a sphere of radius 1075 mi, find an equation for the orbit of Apollo 11. (Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the x-axis.)Plywood Ellipse A carpenter wishes to construct an elliptical table top from a 4 ft by 8 ft sheet of plywood. He will trace out the ellipse using the thumbtack and string method illustrated in Figures 2 and 3. What length of string should he use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?Sunburst Window A sunburst window above a doorway is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 20 in. tall at its highest point and 80 in. wide at the bottom. Find the height of the window 25 in. from the center of the base.56E57E58E59EA hyperbola is the set of all points in the plane for which the __________ of the distances from two fixed points F1 and F2 is constant. The points F1 and F2 are called the __________ of the hyperbola.The graph of the equation x2a2y2b2=1 with a 0, b 0 is a hyperbola with __________ (horizontal/vertical) transverse axis, vertices (___, ___) and (___, ___) and foci (c, 0), where c = __________. So the graph of x242y232=1 is a hyperbola with vertices (___, ___) and (___, ___) and foci (___, ___) and (___, ___).3ELabel the vertices, foci, and asymptotes on the graphs given for the hyperbolas in Exercises 2 and 3. (a) x242y232=1 (b) y242x232=15E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48EComet Trajectories Some comets, such as Halleys comet, are a permanent part of the solar system, traveling in elliptical orbits around the sun. Other comets pass through the solar system only once, following a hyperbolic path with the sun at a focus. The figure below shows the path of such a comet. Find an equation for the path, assuming that the closest the comet comes to the sun is 2 109 mi and that the path the comet was taking before it neared the solar system is at a right angle to the path it continues on after leaving the solar system.Ripples in Pool Two stones are dropped simultaneously into a calm pool of water. The crests of the resulting waves form equally spaced concentric circles, as shown in the figures. The waves interact with each other to create certain interference patterns. (a) Explain why the red dots lie on an ellipse. (b) Explain why the blue dots lie on a hyperbola.51E52E1EThe graphs of x2 = 12y and (x 3)2 = 12(y 1) are given. Label the focus, directrix, and vertex on each parabola.3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41EPath of a Cannonball A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 3200 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.Orbit of a Satellite A satellite is in an elliptical orbit around the earth with the center of the earth at one focus, as shown in the figure. The height of the satellite above the earth varies between 140 mi and 440 mi. Assume that the earth is a sphere with radius 3960 mi. Find an equation for the path of the satellite with the origin at the center of the earth.44ESuppose the x- and y-axes are rotated through an acute angle to produce the new X- and Y-axes. A point P in the plane can be described by its xy-coordinates (x, y ) or its XY-coordinates (X, Y). These coordinates are related by the following formulas. x=X=y=Y=2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE