Question 1091793
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The antenna of a radio telescope is paraboloid measuring 81 feet across with depth of 16 feet. 
Determine, to the nearest tenth of a foot, the distance from the vertex to the focus of this antenna.
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<pre>
If you read the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Parabola-definition--canonical-equation--characteristic-points-and-elements.lesson>Parabola definition, canonical equation, characteristic points and elements</A> 

in this site, you will learn that if you write the canonical equation of a parabola in the form

y = {{{(1/(2*p))*x^2}}},     (1)


then the distance from the parabola's vertex (0,0) to the parabola's focus  (focal distance) is  {{{p/2}}}.


So, our nearest task is to present the equation of our parabola in the form (1).

For it, substitute y = 16 (depth in feet)  and  x = {{{81/2}}}  into equation (1). You will get then

16 = {{{(1/(2*p))*(81/2)^2}}},   or  2p = {{{((81/2)^2)/16}}} = {{{81^2/(4*16)}}}.


Then  p = {{{81^2/(8*16)}}} = 51.258.


Thus the focal distance is  {{{p/2}}} = {{{51.258/2}}} = 25.629 ft.
</pre>

<U>Answer</U>.  The focal distance is  25.629 ft.



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The referred lesson is the part of this online textbook under the topic 
"<U>Conic sections: Parabolas. Definition, major elements and properties. Solved problems</U>".



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