Reflective Property of Parabolas One more day in Sec. 8.1b But first, lets start with a problem: Prove that the graph of the given equation is a parabola, and find its vertex, focus, and directrix. 2 3x 6 x 6 y 10 0 Write in standard form: 3 x 2 2 x 6 y 10 3 x 2 x 1 6 y 10 3 2 2 3 x 1 6 y 7 7
3 x 1 6 y 6 2 7 x 1 2 y 6 2 V: (1, 7/6), F: (1, 5/3), D: y = 2/3 More Practice Prove that the graph of the given equation is a parabola, and find its vertex, focus, and directrix. 2 y 2 y 4 x 12 0 Write in standard form:
2 y 2 y 4 x 12 2 y 2 y 1 4 x 12 1 y 1 2 4 x 13 13 y 1 4 x 4 2 V: (13/4, 1), F: (9/4, 1), D: x = 17/4
More Practice Write an equation for the parabola graphed below. General equation: y k 2 4 p x h Substitute in vertex: (11/2,0) V(1,3) Final Answer: y 3 2 4 p x 1
Substitute in point, solve for p: 0 3 y 3 2 2 4 p 5.5 1 4 p 2 2 x 1 Reflective Property of Parabolas If a parabola is rotated in 3-D space around its axis, the parabola sweeps out a paraboloid of revolution. These are important because a signal (light, sound, radio waves, etc.) emanating from the focus reflects off the
surface of the dish in lines parallel to the axis (and vice versa incoming parallel waves are sent directly toward the focus). Check out Figure 8.9 on p.637!!! Practice Problems During football games, a TV network uses a parabolic reflector with a microphone at the reflectors focus to capture the conversations among players on the field. If the parabolic reflector is 3 ft across and 1 ft deep, where should the microphone be placed? General equation: A diagram of a cross-section 2 x 4 py of the reflector: (1.5, 1)
(1.5, 1) Substitute the points: 2 F(0, p) 1.5 4 p 1 V(0, 0) 2.25 p 0.5625 4 Practice Problems During football games, a TV network uses a parabolic reflector with a microphone at the reflectors focus to capture the conversations among players on the field. If the parabolic
reflector is 3 ft across and 1 ft deep, where should the microphone be placed? A diagram of a cross-section of the reflector: (1.5, 1) (1.5, 1) F(0, p) V(0, 0) Because p = 0.5625 ft, or 6.75 inches, the microphone should be placed inside the reflector along its axis and 6.75 inches from its vertex. Practice Problems The mirror of a flashlight is a paraboloid of revolution. Its diameter is 6 cm and its depth is 2 cm. How far from the vertex should the filament of the light bulb be placed for the flashlight
to have its beam run parallel to the axis of its mirror? Because p = 1.125 cm, the filament should be placed 1.125 cm from the vertex along the axis of the mirror. Practice Problems A main cables of a suspension bridge uniformly distribute the weight of the bridge when in the form of a parabola. The main cables of a particular bridge are attached to towers that are 600 ft apart. The cables are attached to the towers at a height of 110 ft above the roadway and are 10 ft above the roadway at their lowest points. If vertical support cables are at 50-ft intervals along the level roadway, what are the lengths of these vertical cables? Lengths (in feet): 79.444, 54.444, 35, 21.111, 12.778, 10, 12.778, 21.111, 35, 54.444, 79.444
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