document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=749494>Question 749494</A>: <I><SPAN STYLE=\"word-break: break-all;\"> consider the curve y=2logx, where log is the natural logarithm. let &#945; be the tangent to that curve which passes through the origin, let P be the point of contact of &#945; and that curve, and let m be the straight line perpendicular to the tangent &#945; at P. We are to find the equations of the straight lines &#945; and m and the area S of the region bounded by the curve y=2logx, the straight line m, and the x-axis<BR>\n" );
document.write( "let t be the x-coordinate of the poin P, then t satisfies log t=(A). Hence the equation of &#945; is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28B%29%2Ax%2Fe ALIGN=MIDDLE  ALT=\"y=%28B%29%2Ax%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the equation of m is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-ex%2F%28C%29+%2B+e%5E2%2F%28D%29+%2B+%28E%29 ALIGN=MIDDLE  ALT=\"y=-ex%2F%28C%29+%2B+e%5E2%2F%28D%29+%2B+%28E%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "thus the area S of the region is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%28F%29+%2B+%28G%29%2Fe ALIGN=MIDDLE  ALT=\"S=%28F%29+%2B+%28G%29%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "solve for A,B,C,D,E,F and G </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #456151 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=456151\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Aln%28x%29 ALIGN=MIDDLE  ALT=\"y=2%2Aln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Its graph crosses the x-axis at the point where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=2%2Aln%28x%29%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28y=2%2Aln%28x%29%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aln%28x%29=0 ALIGN=MIDDLE  ALT=\"2%2Aln%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28x%29=0 ALIGN=MIDDLE  ALT=\"ln%28x%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The x-coordinate of point P is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t ALIGN=MIDDLE  ALT=\"x=t\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The slope of the tangent at point P is the value of the derivative at that point.<BR>\n" );
document.write( "y'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fx ALIGN=MIDDLE  ALT=\"2%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the slope of the tangent at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t ALIGN=MIDDLE  ALT=\"x=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ft ALIGN=MIDDLE  ALT=\"2%2Ft\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Since the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > tangent at P passes through the origin, its equation must be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2Ft%29x ALIGN=MIDDLE  ALT=\"y=%282%2Ft%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At point P, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t ALIGN=MIDDLE  ALT=\"x=t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2Ft%29t ALIGN=MIDDLE  ALT=\"y=%282%2Ft%29t\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since point P is on the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Aln%28x%29 ALIGN=MIDDLE  ALT=\"y=2%2Aln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, its y-coordinate is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Aln%28t%29 ALIGN=MIDDLE  ALT=\"y=2%2Aln%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=2%2Cy=2%2Aln%28t%29%29 ALIGN=MIDDLE  ALT=\"system%28y=2%2Cy=2%2Aln%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2=2%2Aln%28t%29 ALIGN=MIDDLE  ALT=\"2=2%2Aln%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28t%29=1 ALIGN=MIDDLE  ALT=\"ln%28t%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28highlight%28A=1%29%2Ct=e%29 ALIGN=MIDDLE  ALT=\"system%28highlight%28A=1%29%2Ct=e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and P is (e,2).<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now we can find the equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=%282%2Ft%29x%2Ct=e%29 ALIGN=MIDDLE  ALT=\"system%28y=%282%2Ft%29x%2Ct=e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2Fe%29x ALIGN=MIDDLE  ALT=\"y=%282%2Fe%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28B=2%29%0D%0A+%0D%0ASince+the+slope+of+line+%7B%7B%7Balpha ALIGN=MIDDLE  ALT=\"highlight%28B=2%29%0D%0A+%0D%0ASince+the+slope+of+line+%7B%7B%7Balpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fe ALIGN=MIDDLE  ALT=\"2%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > must have a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F%282%2Fe%29=-e%2F2 ALIGN=MIDDLE  ALT=\"-1%2F%282%2Fe%29=-e%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "As <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > passes through P(e,2) its equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=%28-e%2F2%29%28x-e%29 ALIGN=MIDDLE  ALT=\"y-2=%28-e%2F2%29%28x-e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=%28-e%2F2%29x%2Be%5E2%2F2 ALIGN=MIDDLE  ALT=\"y-2=%28-e%2F2%29x%2Be%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-e%2F2%29x%2Be%5E2%2F2%2B2 ALIGN=MIDDLE  ALT=\"y=%28-e%2F2%29x%2Be%5E2%2F2%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-e%2Ax%2F2%2Be%5E2%2F2%2B2 ALIGN=MIDDLE  ALT=\"y=-e%2Ax%2F2%2Be%5E2%2F2%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28C=2%29 ALIGN=MIDDLE  ALT=\"highlight%28C=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28D=2%29 ALIGN=MIDDLE  ALT=\"highlight%28D=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28E=2%29 ALIGN=MIDDLE  ALT=\"highlight%28E=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > crosses the x-axis at the point where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=-e%2Ax%2F2%2Be%5E2%2F2%2B2%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28y=-e%2Ax%2F2%2Be%5E2%2F2%2B2%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-e%2Ax%2F2%2Be%5E2%2F2%2B2=0 ALIGN=MIDDLE  ALT=\"-e%2Ax%2F2%2Be%5E2%2F2%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-e%2Ax%2Be%5E2%2B4=0 ALIGN=MIDDLE  ALT=\"-e%2Ax%2Be%5E2%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E2%2B4=ex ALIGN=MIDDLE  ALT=\"e%5E2%2B4=ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28e%5E2%2B4%29%2Fe ALIGN=MIDDLE  ALT=\"x=%28e%5E2%2B4%29%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e%2B4%2Fe ALIGN=MIDDLE  ALT=\"x=e%2B4%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The area <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the region bounded by the curve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Aln%28x%29 ALIGN=MIDDLE  ALT=\"y=2%2Aln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the straight line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the x-axis is shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-1%2C6%2C-1%2C6%2C2%2Aln%28x%29%2C-e%2Ax%2F2%2Be%5E2%2F2%2B2%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-1%2C6%2C-1%2C6%2C2%2Aln%28x%29%2C-e%2Ax%2F2%2Be%5E2%2F2%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be can be calculated as the sum of:<BR>\n" );
document.write( "the area below <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Aln%28x%29 ALIGN=MIDDLE  ALT=\"y=2%2Aln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and above the x-axis, between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e ALIGN=MIDDLE  ALT=\"x=e\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%282%2Aln%28x%29%2C+dx%2C+1%2C+e+%29=2%2Aint%28ln%28x%29%2C+dx%2C+1%2C+e+%29 ALIGN=MIDDLE  ALT=\"int%282%2Aln%28x%29%2C+dx%2C+1%2C+e+%29=2%2Aint%28ln%28x%29%2C+dx%2C+1%2C+e+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "plus the area below <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e ALIGN=MIDDLE  ALT=\"x=e\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=e%2B4%2Fe ALIGN=MIDDLE  ALT=\"x=e%2B4%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28-e%2Ax%2F2%2Be%5E2%2F2%2B2%29%2C+dx%2C+e%2C+e%2B4%2Fe+%29 ALIGN=MIDDLE  ALT=\"int%28%28-e%2Ax%2F2%2Be%5E2%2F2%2B2%29%2C+dx%2C+e%2C+e%2B4%2Fe+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28-e%2Ax%2F2%2Be%5E2%2F2%2B2%29%2C+dx%2C+e%2C+e%2B4%2Fe+%29 ALIGN=MIDDLE  ALT=\"int%28%28-e%2Ax%2F2%2Be%5E2%2F2%2B2%29%2C+dx%2C+e%2C+e%2B4%2Fe+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is easier than it seems.<BR>\n" );
document.write( "It's just the area of the triangle with vertices (e,0), P(e,2), and (e+e/4,0)<BR>\n" );
document.write( "Its base is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Fe ALIGN=MIDDLE  ALT=\"4%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >; its height is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and its area is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A2%2A%284%2Fe%29=4%2Fe ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A2%2A%284%2Fe%29=4%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28x%29%2C+dx%29=2%28x%2Aln%28x%29-x%29 ALIGN=MIDDLE  ALT=\"int%28ln%28x%29%2C+dx%29=2%28x%2Aln%28x%29-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Aint%28ln%28x%29%2C+dx%2C+1%2C+e+%29=2%2A%28%28e%2Aln%28e%29-e%29-%281%2Aln%281%29-1%29%29=2%28%28e%2A1-e%29-%281%2A0-1%29%29=2%28%28e-e%29-%280-1%29%29=2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=2%2B4%2Fe ALIGN=MIDDLE  ALT=\"S=2%2B4%2Fe\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28G=4%29 ALIGN=MIDDLE  ALT=\"highlight%28G=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );