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.

Question

​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.

Answer 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