document.write( "Question 1197808: The entrance to a tunnel under a river is in the shape of a parabola. The width of the tunnel at ground level is 20m . At a distance of 4m from one edge of the tunnel, the height is 16m. Find the height of the tunnel in the middle. Please help!!
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #831239 by ikleyn(52800) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "The entrance to a tunnel under a river is in the shape of a parabola.
\n" ); document.write( "The width of the tunnel at ground level is 20m .
\n" ); document.write( "At a distance of 4m from one edge of the tunnel, the height is 16m.
\n" ); document.write( "Find the height of the tunnel in the middle. Please help!!
\n" ); document.write( "~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "There is no need to explain that the parabola is opened down in this problem.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, let assume that the origin of the coordinate system is established at one edge of the tunnel.\r\n" );
document.write( "Then the other edge is at x= 20 meters.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, our parabola has two intersections with x-axis: at x= 0 and x= 20.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus the parabola has two zeroes, 0 and 20. Therefore, we can write it in the form\r\n" );
document.write( "\r\n" );
document.write( "    y = a*(x-0)*(x-20) = a*x*(x-20),\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "where \"a\" is some real coefficient, now unknown.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To find \"a\", use the fact that  y(4) = 16 meters, which is given.  So\r\n" );
document.write( "\r\n" );
document.write( "    a*4*(4-20) = 16,   or\r\n" );
document.write( "\r\n" );
document.write( "    4a*(-16) = 16, \r\n" );
document.write( "\r\n" );
document.write( "    -4a      =  1,\r\n" );
document.write( "\r\n" );
document.write( "      a      = -1/4 = -0.25.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus, the parabola is\r\n" );
document.write( "\r\n" );
document.write( "      y = -0.25x*(x-20).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Its vertex is half-way between 0 and 20, i.e. at x= 10.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, the highest point of the parabola is  y(10) = -0.25*10*(10-20) = -0.25*10*(-10) = 25.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The height of the tunnel is 25 meters over the ground level.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );