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A satellite dish has a shape of paraboloid. The signals that it receives is reflected to the receiver that is located
at the focus of the paraboloid. If dish is 8 feet across at its opening and 1 foot deep at its vertex, determine the location
(distance from the vertex of the dish) of its focus.
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Let us represent the parabola by the equation y = (1).
Then this phrase " . . . dish is 8 feet across at its opening and 1 foot deep at its vertex" means that
the point (x,y) = (4,1) lies on the parabola.
In turn, it then means that y = 1 at x = 4.
It implies that in the equation (1) 1 = , or a = .
So, the equation of the parabola is y = .
It is the canonical equation of the parabola
(see the lesson Parabola definition, canonical equation, characteristic points and elements in this site).
The vertex of this parabola is the point (0,0) (the origin of the coordinate system).
The focus of this parabola is the point ( 0, ) = (0,4).
(See again the lesson referred above).
Thus the distance from the vertex to the focus is 4 feet.
Answer. The distance from the vertex to the focus of the parabola / paraboloid is 4 feet.
Solved.
The solution by @josgarithmetic and his answer "1 feet" is .
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You have this free of charge online textbook in ALGEBRA-II in this site
ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lesson is the part of this online textbook under the topic
"Conic sections: Parabolas. Definition, major elements and properties. Solved problems".