document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143168>Question 1143168</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A square metal plate of side length 2 feet and negligible thickness rests on a table top parallel to the ground, and a point source of light is suspended 2 feet above the center of the plate. While one edge of the plate remains on the table, the opposite edge is lifted until the plate makes a 45∘<BR>\n" );
document.write( " angle with the table top. What is the area, in square feet, of the shadow cast by the plate on the table top </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #764017 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=764017\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Since this question has been sitting here for a couple of days without any response, I will take a stab at it.<br><BR>\n" );
document.write( "There are probably fancy mathematical ways to solve this problem; but I don't know them.<br><BR>\n" );
document.write( "Perhaps another tutor will see my response and either confirm my solution, or show a better method for solving the problem.<br><BR>\n" );
document.write( "So here is my solution....<br><BR>\n" );
document.write( "Picture the metal plate in the xy-plane in a 3D coordinate system, with the center of the plate at the origin.  Have the coordinate system in the usual configuration -- positive z axis vertical up; right-hand rule.<br><BR>\n" );
document.write( "We can label the corners of the plate<BR>\n" );
document.write( "A(1,-1,0)<BR>\n" );
document.write( "B(1,1,0)<BR>\n" );
document.write( "C(-1,1,0)<BR>\n" );
document.write( "D(-1,-1,0)<br><BR>\n" );
document.write( "The point light source (2 feet above the center of the plate) is at P(0,0,2).<br><BR>\n" );
document.write( "Now keep edge AD of the plate on the table and rotate the plate up 45 degrees.  Let B' and C' be the images of B and C after that rotation.<br><BR>\n" );
document.write( "DC' and AB' form 45-45-90 right triangles with the table top; the hypotenuse is 2 and each leg is sqrt(2).  That makes the coordinates of C' (-1,-1+sqrt(2),sqrt(2)).<br><BR>\n" );
document.write( "The edges of the metal plate are straight lines, so the boundaries of the shadow will be straight lines.  So the shaded area will be in the shape of a trapezoid.  And since the light source was centered above the plate, the shape will be an isosceles trapezoid.<br><BR>\n" );
document.write( "The vector PC' is [-1,-1+sqrt(2),sqrt(2)-2].<br><BR>\n" );
document.write( "Let C'' be the projection of PC' onto the table top (z=0).  The vector PC'' is a scalar multiple of the vector PC'; the multiplication factor is the ratio of the differences in the z values, which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%282-sqrt%282%29%29+=+2%2Bsqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2F%282-sqrt%282%29%29+=+2%2Bsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "So vector PC'' is (2+sqrt(2)) times [-1,-1+sqrt(2),sqrt(2)-2] =[-2-sqrt(2),sqrt(2),-2]; and that makes C'' (-2-sqrt(2),sqrt(2),0).<br><BR>\n" );
document.write( "By symmetry, B'' is (2+sqrt(2),sqrt(2),0).<br><BR>\n" );
document.write( "B''C'' is a base of the isosceles trapezoid shaded area; its length is 4+2sqrt(2).<br><BR>\n" );
document.write( "The other base of the shaded area is AD, length 2.<br><BR>\n" );
document.write( "And the height of the pyramid is the difference in the y values of D and C'', which is 1+sqrt(2).<br><BR>\n" );
document.write( "Finally, the area of the isosceles trapezoid shaded area is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%28%28b1%2Bb2%29%2F2%29h+=+%28%282%2B%284%2B2sqrt%282%29%29%29%2F2%29%281%2Bsqrt%282%29%29+=+%283%2Bsqrt%282%29%29%281%2Bsqrt%282%29%29+=+5%2B4sqrt%282%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );