Question 896016
the formula for a parabola that is vertically oriented is:


4p(y-k) = (x-h)^2


since your parabola has the vertex at the origin, then (h,k) = (0,0) which means that h = 0 and k = 0.


the formula therefore becomes:


4py = x^2


p is the distance from the focus to the vertex or the vertex to the directrix, those 2 distances being the same.


since the focus is 5 inches from the vertex, that means that p = 5 and 4p = 20 inches.


the depth of the dish is 2 feet which is equal to 24 inches.


since 4p = 20, the equation of 4py = x^2 becomes:


20y = x^2


divide both sides of this equation by 20 and you get y = x^2/20.


that would be the equation of the parabola in standard quadratic equation form.


since the depth of the antenna is 24 inches, we know that y will be equal to 24 inches and we want to find the points on the parabola at that height.


those points will be the intersection of the parabola with a straight line at y = 24.


since the equation of the parabola is y = x^2 / 20, then we set y = 24 and the equation becomes:


24 = x^2 / 20


multiply both sides of this equation by 20 and you get 480 = x^2.


take the square root of both sides of this equation and you get x = plus or minus sqrt(480) which is roughly equal to plus or minus 21.9089... inches.


the width of the top of the dish is therefore 2 * sqrt(480) which is roughly equal to 43.8178... inches.


the graph of your parabola is shown below.


the focus is at (0,5)
the directrix is the horizontal line at y = -5
the focus and the directrix are both the same distance from the vertex.
that distance is and is equal to 5.
the top of the satellite dish is at y = 24 inches (2 feet).
the points on the parabola when y = 24 are at plus or minus sqrt(480) which is roughly equal to plus or minus 21.9089... inches.
the width of the parabola at that height is 2 * 21.9089... which is roughly equal to 43.8178... inches.


<img src = "http://theo.x10hosting.com/2014/082804.jpg" alt="$$$" </>


two references that talk about parabolas on the web are:


<a href = "http://www.purplemath.com/modules/parabola.htm" target = "_blank">http://www.purplemath.com/modules/parabola.htm</a>
<a href = "http://www.mathsisfun.com/geometry/parabola.html" target = "_blank">http://www.mathsisfun.com/geometry/parabola.html</a>