document.write( "Question 614971: A tunnel through a mountain for a four lane highway is to have a semi-
\n" ); document.write( "elliptical opening. The total width of the highway is 48 ft and the height at
\n" ); document.write( "the edge of the road must be sufficient for a truck 11 ft high to clear. If
\n" ); document.write( "the highest point of the opening is to be 15 ft, approximately how wide must
\n" ); document.write( "the opening be? \n" ); document.write( "

Algebra.Com's Answer #386841 by Edwin McCravy(20062) $\"\"$ $\"About$

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A tunnel through a mountain for a four lane highway is to have a semi-
\n" ); document.write( "elliptical opening. The total width of the highway is 48 ft and the height at
\n" ); document.write( "the edge of the road must be sufficient for a truck 11 ft high to clear. If
\n" ); document.write( "the highest point of the opening is to be 15 ft, approximately how wide must
\n" ); document.write( "the opening be?
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document.write( "The semi-ellipse must be as above.  We put the center at the origin.  Since the\r\n" );
document.write( "height is 15 feet, the semi-minor axis = b = 15.  \r\n" );
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document.write( "The total width of the highway is 48 feet, so each lane must be 24 feet wide.\r\n" );
document.write( "For a truck 11 feet high to just scrape the top of the tunnel, when its\r\n" );
document.write( "right side is at the edge of the highway, the ellipse must pass through the\r\n" );
document.write( "point (24,11), and also the point (-24,11).\r\n" );
document.write( "[Note: It would be more practical to make to make it 1 foot higher, to go\r\n" );
document.write( "through (24,12) instead of (24,11) so that an 11-foot tall truck would have a\r\n" );
document.write( "1-foot clearance and not scrape the top, but the problem didn't state any\r\n" );
document.write( "clearance, so I will let the truck scrape the top at the edge of the highway]\r\n" );
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document.write( "The equation of an ellipse is\r\n" );
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document.write( " +  = 1\r\n" );
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document.write( "But since the center is (h,k) = (0,0), the equation simplifies to\r\n" );
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document.write( " +  = 1\r\n" );
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document.write( "We know that b = 15, so we have\r\n" );
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document.write( " +  = 1\r\n" );
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document.write( " +  = 1\r\n" );
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document.write( "Since it goes through the point (24,11), we substitute (x,y) = (24,11)\r\n" );
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document.write( " +  = 1\r\n" );
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document.write( " +  = 1\r\n" );
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document.write( "Solve that for \"a\" and get\r\n" );
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document.write( "a = 35.30090432, the semi-major axis.\r\n" );
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document.write( "The width of the opening must be the entire major axis,\r\n" );
document.write( "so the answer to your problem is twice that or 70.60180865 feet.\r\n" );
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document.write( "Edwin

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