document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012602>Question 1012602</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ABCD is a square of unit length. Points E and F are on the sides AB and AD respectively such that AE = AF = x. <BR>\n" );
document.write( "Show that the area 'y' of the quadrilateral CDEF is given by y = 1/2 (1 + x - x^2). What is the quadrilaterals greatest possible area. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #628628 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=628628\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> area of triangle EAF = x^2 / 2<BR>\n" );
document.write( "area of triangle EBC = ((1-x)*1) / 2 = (1-x) / 2<BR>\n" );
document.write( "area of square ABCD = 1 * 1 = 1<BR>\n" );
document.write( "area of quadrilateral CDEF(y) = area of square ABCD - area of triangle EAF - area of triangle EBC<BR>\n" );
document.write( "*******************************************************************************<BR>\n" );
document.write( "y = 1 - x^2/2 - (1-x) / 2<BR>\n" );
document.write( "y = (2 - x^2 - (1-x)) / 2<BR>\n" );
document.write( "y = (1 +x -x^2) / 2<BR>\n" );
document.write( "******************************************************************************<BR>\n" );
document.write( "this is a parabola that opens downward, so we want the value of x at the parabola's peak.<BR>\n" );
document.write( "x = -b / 2a = (-1/2) / (2 * (-1/2)) = 1/2<BR>\n" );
document.write( "now substitute 1/2 for x in the equation for y<BR>\n" );
document.write( "y = (1 +(1/2) - (1/2)^2) / 2 = (5/4) / 2 = 5/8<BR>\n" );
document.write( "******************************************************************************<BR>\n" );
document.write( "the greatest possible area for quadrilateral CDEF is 5/8<BR>\n" );
document.write( "******************************************************************************<BR>\n" );
document.write( "here is the graph of the area of quadrilateral CDEF<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C+200%2C+-3%2C+3%2C+-3%2C+3%2C+%281+%2Bx+-x%5E2%29+%2F+2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C+200%2C+-3%2C+3%2C+-3%2C+3%2C+%281+%2Bx+-x%5E2%29+%2F+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );