- What is the mathematical definition of this conic section and how does that definition play a role in the properties of the conic section and how it is shaped or formed? The definition of a conic section is the figure(s) formed by the intersection of a plane through two double napped cones. The figures that can be cut are circles, ellipses,parabolas, and hyperbolas. In order to make a circle the plane must cut horizontally perpendicular to one cone. To make an ellipse it must be cut less than 90 degrees of the cone. In making a parabola, the plane must cut the cone parallel to one of the slanted sides of the cone. Lastly to make a hyperbola, the plane must cut both cones perpendicularly to the base to forms a hyperbola.
- How does the focus (or foci) affect the shape of the conic section? In a parabola, the distance between the vertex and the focus is equal to 'p', similarly the distance between the vertex and the directrix equals 'p'. So, as the parabola goes in what ever direction it does, according to the standard form, the distance between the focus and any point on the graph is equal to the distance between that point and a distance perpendicular to the directrix. .
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- How do the properties of this conic section apply in real life? Whispering dishes are shaped as a paraboloid, which makes it possible to whisper into it and have it heard from across a room. What makes this possible is the properties of a parabola, as you say soemthing in one dish, the sound waves bounce off the edges and then hits the other dish which focuses the sound into the focal point (focus). Citations:
Video for Conic Sections: http://www.youtube.com/watch?v=BnWEI5o35G8
Parabola Picture: http://www.personal.kent.edu/~rmuhamma/Algorithms/MyAlgorithms/Gifs/parabola-2.gif
Whispering Dishes Video:http://www.youtube.com/watch?feature=player_embedded&v=y26wYeOg_gU
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