Question 1184546


A radio telescope has a parabolic dish with a diameter of {{{110}}}  meters. The collected radio signals are reflected to one collection point, called the focal point, being the focus of the parabola. 
If the focal length is {{{50}}} meters, find the depth of the dish, rounded to one decimal place.

To simplify my computations, I'll put the vertex of my parabola (that is, the base of the dish) at the origin, so ({{{h}}}, {{{k}}}) = ({{{0}}}, {{{0}}}). 

{{{4py = x^2}}}

Since the focal length is {{{50}}}, then {{{p = 50}}} and the equation is:

{{{4(50)y = x^2}}}
{{{200y = x^2}}}
{{{y = x^2/200}}}


This parabola extends forever in either direction, but I only care about the part of the curve that models the dish. Since the dish has a diameter of a {{{110}}} meters, then I only care about the part of the curve from {{{x = -55 }}}to {{{x = +55}}}.

The height of the edge of the dish (and thus the depth of the dish) will be the y-value of the equation at the "ends" of the modelling curve. The height of the parabola will be the same at either x-value, since they're each the same distance from the {{{vertex}}}, so it doesn't matter which value I use. I prefer positive values, so I'll plug {{{x = 55 }}}into my modelling equation:

{{{y = (55)^2/200}}}

{{{ y = 3025/200 }}}
 
{{{y=121/8}}} or about {{{15.13}}} meters