SECTION 10.3 The Ellipse 703 hole in his roof to accommodate the pipe. If the pitch of his roof is 5 4 (a rise of 5, run of 4), what are the dimensions of the hole? Source: www.doe.virginia.gov 80. Volume of a Football A football is in the shape of a prolate spheroid, which is simply a solid obtained by rotating an ellipse x a y b 1 2 2 2 2 ( ) + = about its major axis.An inflated NFL football averages 11.125 inches in length and 28.25 inches in center circumference. If the volume of a prolate spheroid is ab 4 3 ,2 π how much air does the football contain? (Neglect material thickness.) Source: nfl.com In Problems 81–85, use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus.The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. See the figure. Major axis Sun Center Aphelion Mean distance Perihelion 81. Earth The mean distance of Earth from the Sun is 93 million miles. If the aphelion of Earth is 94.5 million miles, what is the perihelion? Find an equation for the orbit of Earth around the Sun. 82. Mars The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is 128.5 million miles, what is the aphelion? Find an equation for the orbit of Mars about the Sun. 83. Jupiter The aphelion of Jupiter is 507 million miles. If the distance from the center of its elliptical orbit to the Sun is 23.2 million miles, what is the perihelion? What is the mean distance? Find an equation for the orbit of Jupiter around the Sun. 84. Pluto The perihelion of Pluto is 4551 million miles, and the distance from the center of its elliptical orbit to the Sun is 897.5 million miles. Find the aphelion of Pluto. What is the mean distance of Pluto from the Sun? Find an equation for the orbit of Pluto about the Sun. 85. Elliptical Orbit A planet orbits a star in an elliptical orbit with the star located at one focus. The perihelion of the planet is 5 million miles. The eccentricity e of a conic section is e c a . = If the eccentricity of the orbit is 0.75, find the aphelion of the planet.† 86. A rectangle is inscribed in an ellipse with major axis of length 14 meters and minor axis of length 4 meters. Find the maximum area of a rectangle inscribed in the ellipse. Round your answer to two decimal places.† 87. Lithotripsy Extracorporeal shock wave lithotripsy is a procedure that uses shockwaves to fragment kidney stones without the need for surgery. Using an elliptical reflector, a shock wave generator is placed at one focus and the kidney stone is positioned at the other focus. Shock waves pass through a water column and disintegrate the stone, allowing it to be passed out of the body naturally. 71. Semielliptical Arch Bridge An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river 20 meters wide.The center of the arch is 6 meters above the center of the river. See the figure. Find an equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch. 6 m 20 m 72. Semielliptical Arch Bridge The arch of a bridge is a semiellipse with a horizontal major axis. The span is 30 feet, and the top of the arch is 10 feet above the major axis. The roadway is horizontal and is 2 feet above the top of the arch. Find the vertical distance from the roadway to the arch at 5-foot intervals along the roadway. 73. Whispering Gallery A hall 100 feet in length is to be designed as a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center? 74. Whispering Gallery Jim, standing at one focus of a whispering gallery, is 6 feet from the nearest wall. His friend is standing at the other focus, 100 feet away. What is the length of this whispering gallery? How high is its elliptical ceiling at the center? 75. Semielliptical Arch Bridge A bridge is built in the shape of a semielliptical arch. The arch has a span of 120 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the height of the arch at distances of 10, 30, and 50 feet from the center. 76. Semielliptical Arch Bridge A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100 feet. The height of the arch, at a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center. 77. Racetrack Design Consult the figure. A racetrack is in the shape of an ellipse 100 feet long and 50 feet wide.What is the width 10 feet from a vertex? 100 ft 50 ft ? 10 ft 78. Semielliptical Arch Bridge An arch for a bridge over a highway is in the form of half an ellipse.The top of the arch is 20 feet above the ground level (the major axis).The highway has four lanes, each 12 feet wide; a center safety strip 8 feet wide; and two side strips, each 4 feet wide. What should the span of the arch be (the length of its major axis) if the height 28 feet from the center is to be 13 feet? 79. Installing a Vent Pipe A homeowner is putting in a fireplace that has a 4-inch-radius vent pipe. He needs to cut an elliptical †Courtesy of the Joliet Junior College Mathematics Department Applications and Extensions
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