Question 896016: I need help determining the width of a dish (satellite) if the depth is 2 feet and the focus is 5 inches from the vertex. Assuming that the vertex is at (0,0).
Found 2 solutions by Theo, lwsshak3:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
two references that talk about parabolas on the web are:
http://www.purplemath.com/modules/parabola.htm
http://www.mathsisfun.com/geometry/parabola.html
Answer by lwsshak3(11628)  (Show Source):
You can put this solution on YOUR website! need help determining the width of a dish (satellite) if the depth is 2 feet and the focus is 5 inches from the vertex. Assuming that the vertex is at (0,0).
***
Draw a parabola that opens up with vertex at the origin.
Its basic equation: x^2=4py
(x,2)=(x,y)coordinates of top right edge of dish
p=5/12 (distance from vertex to focus)
4p=5/3
using coordinates of top right edge of dish
equation: x^2=(5/3)2
x=√(10/3)
2x≈3.65
width of dish≈3.65 ft
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