document.write( "Question 1199585: The towers of a 60 meter parabolic suspension bridge are 12 m high and the lowest point of the cable is 3 m above the roadway. Find the vertical distance from the roadway to the cable at 15 m from the center.
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Algebra.Com's Answer #833533 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "The towers of a 60 meter parabolic suspension bridge are 12 m high and the lowest point
\n" ); document.write( "of the cable is 3 m above the roadway. Find the vertical distance from the roadway
\n" ); document.write( "to the cable at 15 m from the center.
\n" ); document.write( "A. 3 m
\n" ); document.write( "B. 6 m
\n" ); document.write( "C. 5 m
\n" ); document.write( "D. 8 m
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document.write( "If we place the origin of the coordinate system at the bridge level midpoint between the two towers, \r\n" );
document.write( "we have the vertex of the parabola at the point (0,3).\r\n" );
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document.write( "So, we write an equation of the parabola in vertex form\r\n" );
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document.write( "    y = ax^2 + 3.\r\n" );
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document.write( "Coefficient \"a\" is unknown.  It is the only unknown in this problem now.\r\n" );
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document.write( "To find it, we use the condition at the endpoint: y= 12 at x= 60/2 = 30.  It gives\r\n" );
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document.write( "    12 = a*30^2 + 3\r\n" );
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document.write( "    12 - 3 = a*900\r\n" );
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document.write( "       9   = 900a\r\n" );
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document.write( "        a   =  = .\r\n" );
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document.write( "Thus the parabola is  y = .    \r\n" );
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document.write( "Having this equation ready, we substitute x =15 meters into the equation \r\n" );
document.write( "and find the height of the cable at the point x= 15, which is 15 meters from the midpoint of the bridge\r\n" );
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document.write( "    y =  = 2.25 + 3 = 5.25 m   (rounded).    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "The problem can be solved mentally, without using equations, but using common sense, instead.\r
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\n" ); document.write( "\n" ); document.write( "Half-length of the bridge between the towers is 60/2 = 30 m.\r
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\n" ); document.write( "\n" ); document.write( "The point of the interest is at 15 m, half of the bridge's half-length, from the midpoint.\r
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\n" ); document.write( "\n" ); document.write( "So, the height of the cable above the bridge is 3 m   PLUS   1/4 of the cable levels difference between
\n" ); document.write( "the midpoint position and the tower position\r
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\n" ); document.write( "\n" ); document.write( "     height = 3 + 1/4*(12-3) = 3 + 1/4*9 = 3 + 2.25 = 5.25 m,\r
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\n" ); document.write( "\n" ); document.write( "exactly the same as we obtained it above.\r
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\n" ); document.write( "\n" ); document.write( "This value   5.25 m   is close to the value 5 of the answer option C.\r
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