700 CHAPTER 10 Analytic Geometry Figure 32(a) Elliptical orbits Asteroids Mars Earth Venus Jupiter A Whispering Gallery The whispering gallery in the Museum of Science and Industry in Chicago can be modeled by the top half of a three-dimensional ellipse, which is called an ellipsoid. The gallery is 47.3 feet long. The distance from the center of the room to the foci is 20.3 feet. Find an equation that describes the shape of the room. How high is the room at its center? Source: Chicago Museum of Science and Industry website; www.msichicago.org Solution EXAMPLE 9 Set up a rectangular coordinate system so that the center of the ellipse is at the origin and the major axis is along the x-axis. The equation of the ellipse is x a y b 1 2 2 2 2 + = Since the length of the room is 47.3 feet, the distance from the center of the room to each vertex (the end of the room) will be 47.3 2 23.65 = feet; so a 23.65 feet. = The distance from the center of the room to each focus is c 20.3 feet. = See Figure 33. Because b a c , 2 2 2 = − this means that b 23.65 20.3 147.2325. 2 2 2 = − = An equation that describes the shape of the room is given by x y 23.65 147.2325 1 2 2 2 + = The height of the room at its center is b 147.2325 12.1 feet. = ≈ Figure 33 Whispering gallery model y 15 x 25 225 (20.3, 0) (23.65, 0) (223.65, 0) (220.3, 0) (0, 12.1) 3 Solve Applied Problems Involving Ellipses Ellipses are found in many applications in science and engineering. For example, the orbits of the planets around the Sun are elliptical, with the Sun’s position at a focus. See Figure 32(a). Stone and concrete bridges are often shaped as semielliptical arches. Elliptical gears are used in machinery when a variable rate of motion is required. Ellipses also have an interesting reflection property. If a source of light (or sound) is placed at one focus, the waves transmitted by the source will reflect off the ellipse and concentrate at the other focus. See Figure 32(b). This is the principle behind whispering galleries, which are rooms designed with elliptical ceilings. A person standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus, because all the sound waves that reach the ceiling are reflected to the other person. Now Work PROBLEM 73 Figure 32(b) Reflective property Focus Focus Credit: J. B. Spector/Museum of Science and Industry, Chicago/Getty Images
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