document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186192>Question 1186192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Mrs. Flores recently subscribed in a tv cable plan at GSAT Company. The<BR>\n" );
document.write( "receiving dish of the GSAT Cable is in the shape of a paraboloid of revolution.<BR>\n" );
document.write( "Find the location of the receiver which is placed at the focus if the dish is 12<BR>\n" );
document.write( "feet across and 3 feet deep. Also write the standard form equation of the ellipse. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849472 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849472\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the location of the receiver and the equation of the paraboloid:\r<BR>\n" );
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document.write( "**1. Set up a coordinate system:**\r<BR>\n" );
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document.write( "It's easiest to place the vertex of the paraboloid at the origin (0,0) and have the parabola open upwards.  This makes the axis of symmetry along the y-axis.\r<BR>\n" );
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document.write( "**2. Identify key points:**\r<BR>\n" );
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document.write( "*   The dish is 12 feet across, so it extends 6 feet to either side of the y-axis.  This gives us two points on the parabola: (-6, 3) and (6, 3).<BR>\n" );
document.write( "*   The dish is 3 feet deep, so the vertex is at (0,0) and the focus is somewhere along the positive y-axis.\r<BR>\n" );
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document.write( "**3. Standard equation of a parabola:**\r<BR>\n" );
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document.write( "The standard form equation of a parabola opening upwards with its vertex at the origin is:\r<BR>\n" );
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document.write( "x² = 4py\r<BR>\n" );
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document.write( "where 'p' is the distance from the vertex to the focus.\r<BR>\n" );
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document.write( "**4. Solve for 'p' (the distance to the focus):**\r<BR>\n" );
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document.write( "We can use one of the points we identified, such as (6, 3), and plug it into the equation to solve for 'p':\r<BR>\n" );
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document.write( "6² = 4p * 3<BR>\n" );
document.write( "36 = 12p<BR>\n" );
document.write( "p = 3\r<BR>\n" );
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document.write( "**5. Location of the receiver (focus):**\r<BR>\n" );
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document.write( "Since the vertex is at (0,0) and p = 3, the focus is located at (0, 3).  This means the receiver should be placed 3 feet above the vertex of the dish.\r<BR>\n" );
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document.write( "**6. Standard form equation of the paraboloid:**\r<BR>\n" );
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document.write( "Now that we know p = 3, we can plug it into the standard equation:\r<BR>\n" );
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document.write( "x² = 4 * 3 * y<BR>\n" );
document.write( "x² = 12y\r<BR>\n" );
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document.write( "Therefore, the location of the receiver (focus) is **3 feet above the vertex**, and the standard form equation of the paraboloid is **x² = 12y**.<BR>\n" );
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