document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197396>Question 1197396</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rectangular piece of metal is 15 in longer than it is wide. Squares with sides 3 in long are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 1092 in​^3, what were the original dimensions of the piece of​ metal? <BR>\n" );
document.write( "What is the original​ width? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830645 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=830645\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "x = width of sheet<BR>\n" );
document.write( "x+15 = length<br><BR>\n" );
document.write( "After cutting the 3-inch squares out of the corners and folding the flaps up, the dimensions of the open box are<BR>\n" );
document.write( "width: x-2(3) = x-6<BR>\n" );
document.write( "length: (x+15)-2(3) = x+9<BR>\n" );
document.write( "depth: 3<br><BR>\n" );
document.write( "The volume is 1092 cubic inches:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x-6%29%28x%2B9%29=1092 ALIGN=MIDDLE  ALT=\"3%28x-6%29%28x%2B9%29=1092\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28x%5E2%2B3x-54%29=1092 ALIGN=MIDDLE  ALT=\"3%28x%5E2%2B3x-54%29=1092\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x-54=364 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x-54=364\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B3x-418=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B3x-418=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "It looks as if that quadratic is going to be hard to factor, so we could use the quadratic formula.  However, if the problem is well designed, the answer should be an integer, so let's try to do the factoring.<br><BR>\n" );
document.write( "We need two numbers whose product is 418 and whose difference is 3.<br><BR>\n" );
document.write( "418 = 2(209) = 2(11)(19) = (22)(19)<br><BR>\n" );
document.write( "So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B22%29%28x-19%29=0 ALIGN=MIDDLE  ALT=\"x%2B22%29%28x-19%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-22 ALIGN=MIDDLE  ALT=\"x=-22\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=19 ALIGN=MIDDLE  ALT=\"x=19\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Clearly the negative answer makes no sense in the problem, so<br><BR>\n" );
document.write( "ANSWER: the original width is x = 19 inches<br><BR>\n" );
document.write( "CHECK: volume = 3(x-6)(x+9) = 3(13)(28) = 1092<br><BR>\n" );
document.write( "Note that if a formal algebraic solution is not required, you can solve the problem with a bit of logical reasoning and some simple arithmetic.<br><BR>\n" );
document.write( "The volume is 1092, and the depth is 3; so length times width is 1092/3 = 364.<br><BR>\n" );
document.write( "So now we need two integers whose difference is 15 and whose product is 364:<br><BR>\n" );
document.write( "364 = 4(91) = 4(7)(13) = 28*13<br><BR>\n" );
document.write( "The width of the box is 13 inches; so before the 3-inch squares were cut out of the corners, the original width was 13+6 = 19 inches.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );