Question

the reflecting dish of a parabolic microphone has a cross-section in the shape of a parabola.the microphone itself is placed on the focus of the parabola. if the parabola is 24 inches wide and 4 inches deep,how far from the vertex should the microphone be placed?

Answer 1

Well we could look at it like this. we can make a generic equation for this by saying that the vertex lies at (0,0). we know that it is 24 inches wide. this means that on either side of the vertex is 12 inches. since it is 4 inches deep, we know that these points at x = -12, 12 have a y = 4. to make the equation we look at the vertex form of parabolas:
y=a(x−h)^2+k

we can plug in the point (-12,4) and the vertex
4 = a (-12 + 0)^2 + 0
4 = a (144)
a = 1/36

the vertex is
(h,k) or (0,0)

and the focus point :
(h,k+1/4a) or (0,0 + 1/(4/32)
(0, 1/(1/9))
(0, 9)

this would mean that the focus point is 9 inches above the vertex.