Algebra

An engineer designs a satellite dish with a parabolic cross section. The dish is 10 ft wide at the​ opening, and the focus is placed 8 ft from the vertex.

​a) Position a coordinate system with the origin
at the vertex and the​ x-axis on the​ parabola's
axis of symmetry and find an equation of the
parabola.

​b) Find the depth of the satellite dish at the vertex.

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  1. place the vertex at (0,0) and the focus at (8,0)
    The directrix will be x = -8

    let P(x,y) be any point
    By definition:
    √( (x-8)^2 + y^2) = x+8
    square both sides
    x^2 - 16x + 64 + y^2 = x^2 + 16x + 64

    y^2 = 32x

    so at the point where the dish is 10 ft wide, y = 5
    25 = 32x
    x = 25/32

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