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Question 1187286: Sound technicians at professional sports events often use parabolic receivers as they move along sidelines. If two -dimensional cross section of the receiver is modeled by the equation y2= 55x and is 36 inches in diameter, how deep is the parabolic receiver? What is the location of the focus?
(1) Given
(2) Required to find
(3) Graph/Illustration
(4) Equation to be used
(5) Computation
(6) Final answer
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! (1) Given
 <-->  <--> 
(2) Required to find: Depth (we need to visualize w/graph)
(3) Graph/Illustration
The rectangle represents a cross-section of half the parabolic receiver
(4) Equation to be used
The diameter of the edge of that parabolic dish is 30in, so the radius is 
To find the depth, we need the value of  for  , so we use 
To find the location of the focus, we either use a formula, or we apply the definition of parabola.
If you were told that a parabola with the x-axis for an axis of symmetry and the focus at  has the equation  That is the equation to be used
The focus is at a point and the directrix is the line 
The point  in the parabola with  is at a distance  from the directrix.
By definition, it must also be at a the same distance  from .
Since  is in the parabola with  and  ,
we can use applied to to find 
(5) Computation
Depth:
For  in inches, or  or 
Focus location:
From comparing with we get
 -->  --> 
(6) Final answer
The depth of the parabolic receiver is  ,
and the focus of the parabolic receiver is  in front of the center of the parabolic dish, along its axis of symmetry.
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