document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=896016>Question 896016</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help determining the width of a dish (satellite) if the depth is 2 feet and the focus is 5 inches from the vertex. Assuming that the vertex is at (0,0).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #543187 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=543187\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula for a parabola that is vertically oriented is:\r<BR>\n" );
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document.write( "4p(y-k) = (x-h)^2\r<BR>\n" );
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document.write( "since your parabola has the vertex at the origin, then (h,k) = (0,0) which means that h = 0 and k = 0.\r<BR>\n" );
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document.write( "the formula therefore becomes:\r<BR>\n" );
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document.write( "4py = x^2\r<BR>\n" );
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document.write( "p is the distance from the focus to the vertex or the vertex to the directrix, those 2 distances being the same.\r<BR>\n" );
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document.write( "since the focus is 5 inches from the vertex, that means that p = 5 and 4p = 20 inches.\r<BR>\n" );
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document.write( "the depth of the dish is 2 feet which is equal to 24 inches.\r<BR>\n" );
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document.write( "since 4p = 20, the equation of 4py = x^2 becomes:\r<BR>\n" );
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document.write( "20y = x^2\r<BR>\n" );
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document.write( "divide both sides of this equation by 20 and you get y = x^2/20.\r<BR>\n" );
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document.write( "that would be the equation of the parabola in standard quadratic equation form.\r<BR>\n" );
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document.write( "since the depth of the antenna is 24 inches, we know that y will be equal to 24 inches and we want to find the points on the parabola at that height.\r<BR>\n" );
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document.write( "those points will be the intersection of the parabola with a straight line at y = 24.\r<BR>\n" );
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document.write( "since the equation of the parabola is y = x^2 / 20, then we set y = 24 and the equation becomes:\r<BR>\n" );
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document.write( "24 = x^2 / 20\r<BR>\n" );
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document.write( "multiply both sides of this equation by 20 and you get 480 = x^2.\r<BR>\n" );
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document.write( "take the square root of both sides of this equation and you get x = plus or minus sqrt(480) which is roughly equal to plus or minus 21.9089... inches.\r<BR>\n" );
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document.write( "the width of the top of the dish is therefore 2 * sqrt(480) which is roughly equal to 43.8178... inches.\r<BR>\n" );
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document.write( "the graph of your parabola is shown below.\r<BR>\n" );
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document.write( "the focus is at (0,5)<BR>\n" );
document.write( "the directrix is the horizontal line at y = -5<BR>\n" );
document.write( "the focus and the directrix are both the same distance from the vertex.<BR>\n" );
document.write( "that distance is and is equal to 5.<BR>\n" );
document.write( "the top of the satellite dish is at y = 24 inches (2 feet).<BR>\n" );
document.write( "the points on the parabola when y = 24 are at plus or minus sqrt(480) which is roughly equal to plus or minus 21.9089... inches.<BR>\n" );
document.write( "the width of the parabola at that height is 2 * 21.9089... which is roughly equal to 43.8178... inches.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/082804.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "two references that talk about parabolas on the web are:\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/parabola.htm\" target = \"_blank\">http://www.purplemath.com/modules/parabola.htm</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.mathsisfun.com/geometry/parabola.html\" target = \"_blank\">http://www.mathsisfun.com/geometry/parabola.html</a>\r<BR>\n" );
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