A satellite dish antenna is to be constructed in the shape of a paraboloid. The paraboloid is formed by rotating the parabola with focus at the point (25, 0) and directrix x=−25 about the x-axis, where x and y are in inches. The diameter of the antenna is to be 80 inches. Find the equation of the parabola
Answers
Answer:
A 25,0 and directed -25 the equation is 35 inches
Answer:
Focus of a Parabola
We first write the equations of the parabola so that the focal distance (distance from vertex to focus) appears in the equation.
The figure below shows a parabola, its focus F at (0,f) and its directrix at y = -f.
Parabolic dish
We now use the definition of the parabola. Any point M(x,y) on the parabola is equidistant from the focus and the directrix. Hence the equation:
sqrt[(x-0)2 + (y - f)2] = sqrt[(x - x)2 + (y - (-f))2]
We now square both sides and expand the squares.
x2 + y2 + f2 - 2y*f = y2 + f2 + 2y*f
Simplify to obtain the equation of the parabola involving the focal distance f.
y = x2 / 4f
We now look at a more practical situation where we know the dimensions of the dish and we want to find the focal distance which gives the position of the focus relative to the position of the dish as swown in the figure below.
Parabolic dish with dimensions: diameter and depth
D is the diameter of the dish, d is the depth of the dish and f is the focal distance.
The points (D/2,d) and (-D/2,d) are on the parabola, hence
d = (D/2)2 / 4f
Which gives a relationship between the diameter D, the depth d and the focal distance f of the dish.
f = D2 / 16d
Step-by-step explanation:
The above formula helps in positioning the feed of the parabolic antennas as it gives the focal distance f. Of course in practice the shape of the dish is not a perfect parabola and therefore small adjustments are needed when positioning the feed.
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