Reflective Property of Parabolas One more day in

Reflective Property of Parabolas One more day in Sec. 8. 1 b…

But first, let’s start with a problem: Prove that the graph of the given equation is a parabola, and find its vertex, focus, and directrix. Write in standard form: 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. Write in standard form: V: (13/4, 1), F: (9/4, 1), D: x = 17/4

More Practice Write an equation for the parabola graphed below. General equation: Substitute in vertex: (11/2, 0) V(1, – 3) Final Answer: Substitute in point, solve for p:

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 reflector’s 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 of the reflector: (– 1. 5, 1) (1. 5, 1) F(0, p) V(0, 0) Substitute the points:

Practice Problems During football games, a TV network uses a parabolic reflector with a microphone at the reflector’s 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