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Need some help with precalc?
a satellite dish is shaped like a paraboloid of revolution. the signals that emanate form a sattelite strike the surface of the dish and are reflected to a single point, where the receive is located.if the dish is 14 feet across at its opening and 4 feet deep at its center, at what position should the receiver be placed.
2 Answers
- BrianLv 78 years agoFavorite Answer
The standard form equation for a parabola which opens upward is
y = (1/(4p))*(x - h)^2 + k,
where p is the focal length and (h,k) is the vertex.
The receiver will need to be placed at the focal point, i.e., a distance of p
directly above the vertex of the dish.
Look at a mid-cross-section of the paraboloid, and let the vertex
of the dish be at (0,0). Then the points (-7,4) and (7,4) will also lie on this
cross-section. So plugging x = 7, y = 4 and (h,k) = (0,0) into our formula, we have
4 = (1/(4p))*7^2 ----> 4p = 49/4 ------> p = 49/16.
So the receiver need to be placed (49/16) feet directly above the vertex of the dish.
- DWReadLv 78 years ago
The receiver is placed at the focus.
If the vertex is at the origin, then (0, 0), (-7, 4), and (7, 4) are points on the parabola.
Equation of the parabola:
y = ax²
Use (7, 4) to determine a.
4 = a7²
a = 4/49
focal length p = 1/(4a) = 49/16
focus (0, 49/16)
The focus should be placed 3 1/16 feet above the center.
