document.write( "Question 1171631: A truck is about to pass through a one-way tunnel in the form of a semi-ellipse, which 15 m across and 4 m high in the middle. If the truck has a width of 4 m and a height of 3.5 mwill be able to pass through the tunnel ? Justify your answer . \n" ); document.write( "

Algebra.Com's Answer #796559 by htmentor(1343) $\"\"$ $\"About$

You can put this solution on YOUR website!
Standard form for an ellipse centered at the origin is:
\n" ); document.write( "(x/a)^2 + (y/b)^2 = 1, where a, b are the semi-major and semi-minor
\n" ); document.write( "axes, respectively. Since the width of the tunnel = 15, a = 7.5.
\n" ); document.write( "The height = b = 4.
\n" ); document.write( "Thus the equation describing the tunnel opening is:
\n" ); document.write( "(x/7.5)^2 + (y/4)^2 = 1
\n" ); document.write( "A truck having width 4 and height 3.5 will be able to pass through
\n" ); document.write( "the tunnel. We can see this mathematically and graphically.
\n" ); document.write( "Mathematically, we only need to consider if the sides of the
\n" ); document.write( "truck clear the topof the tunnel, since that would be the
\n" ); document.write( "lowest point. The sides of the truck are at x = +-2.
\n" ); document.write( "We need the height of the tunnel at these points:
\n" ); document.write( "(y/4)^2 = 1 - (2/7.5)^2 -> y = 4*sqrt(1-(2/7.5)^2) = 3.855 m.
\n" ); document.write( "Since this is greater than the truck height, the truck can pass through.
\n" ); document.write( "We can also see this graphically:
\n" ); document.write( "

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

\n" );