document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=495218>Question 495218</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve the word problem using<BR>\n" );
document.write( "The Rational Zero Theorem\r<BR>\n" );
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document.write( "An open metal tank is to be made from a rectangular piece of stainless steel that measures 12 by 6 feet, by cutting out squares of the same size from each corner and bending up the sides. If the volume of the tank is to be 50 ft^3 , how large the square should be cut from each corner? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #336020 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=336020\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> well, i didn't solve your problem for you, although, in a sense i did.<BR>\n" );
document.write( "the reason is that the problem doesn't have a solution for any of the dimensions being all greater than 0.<BR>\n" );
document.write( "here's what i did.<BR>\n" );
document.write( "your piece of steel has a length of 12 feet and a width of 6 feet.<BR>\n" );
document.write( "you want to cut off a square from each end and then fold up the sides to make a tank that will have a volume of 50 cubic feet.<BR>\n" );
document.write( "your tank will have a height, a length, and a width.<BR>\n" );
document.write( "we will let x represent the height.<BR>\n" );
document.write( "x also represents the side of the square that will be removed from each corner of your rectangle.<BR>\n" );
document.write( "the length of the tank will be represented by 12 - 2x because x is cut off the length from both ends.<BR>\n" );
document.write( "the width of the tank will be represented by 6 - 2x because x is cut off the width from both ends.<BR>\n" );
document.write( "the picture below shows you what i mean.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2011/sep111.jpg\" alt = \"$$$$$\" / ><BR>\n" );
document.write( "before going through the drill of trying to find the factors, i drew a graph of the equation that gave me the volume of the tank.<BR>\n" );
document.write( "the formula for that volume is:<BR>\n" );
document.write( "L * W * H = V<BR>\n" );
document.write( "V = volume<BR>\n" );
document.write( "L = length<BR>\n" );
document.write( "W = width<BR>\n" );
document.write( "H = height.<BR>\n" );
document.write( "the height is equal to x.<BR>\n" );
document.write( "the length is equal to 12 - x<BR>\n" );
document.write( "the width is equal to 6 - x<BR>\n" );
document.write( "the equation that i got was:<BR>\n" );
document.write( "V = L * W * H which became:<BR>\n" );
document.write( "V = (12 - 2x) * (6 - 2x) * x<BR>\n" );
document.write( "since V = 50, this equation became:<BR>\n" );
document.write( "(12 - 2x) * (6 - 2x) * x = 50<BR>\n" );
document.write( "to graph this equation, i subtracted 50 from both sides of the equation and then set the equation equal to y to get:<BR>\n" );
document.write( "y = (12 - 2x) * (6 - 2x) * x - 50<BR>\n" );
document.write( "when y = 0, i have a solution to this equation.<BR>\n" );
document.write( "when y = 0 this equation becomes:<BR>\n" );
document.write( "(12 - 2x) * (6 - 2x) * x - 50 = 0<BR>\n" );
document.write( "if you add 50 to both sides of this equation, then you get:<BR>\n" );
document.write( "(12 - 2x) * (6 - 2x) * x = 50<BR>\n" );
document.write( "this is exactly the equation for the volume of the tank.<BR>\n" );
document.write( "so, when y = 0, this equation has a solution for the volume of the tank.<BR>\n" );
document.write( "to see when y = 0, i graphed the equation of y = (12 - 2x) * (6 - 2x) * x - 50.<BR>\n" );
document.write( "the graph that i got is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2011/sep112.jpg\" alt = \"$$$$$\" / ><BR>\n" );
document.write( "while this provides a solution, the solution is not valid.<BR>\n" );
document.write( "it is not valid because:<BR>\n" );
document.write( "when x = 6.5:<BR>\n" );
document.write( "L = (12 - 2x) = (12 - 13) = -1<BR>\n" );
document.write( "W = (6 - 2x) = (6 - 13) = -7<BR>\n" );
document.write( "H = 6.5<BR>\n" );
document.write( "V = 6.5 * (-1) * (-7) = 45.5<BR>\n" );
document.write( "the actual value of x that gets us close to a volume of 50 is 6.53993479<BR>\n" );
document.write( "that doesn't matter.<BR>\n" );
document.write( "what matters is that the dimensions of the tank are negative which can't be.<BR>\n" );
document.write( "what this means is we don't have a solution to this problem where the length and the width and the height are all positive.<BR>\n" );
document.write( "so the answer to the problem is that there is no solution that is possible based on the requirements of the equation.<BR>\n" );
document.write( "you can have a tank with a volume of 50 cubic feet.<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "a length of 5 and a width of 5 and a height of 2 will get you a tank with a volume of 50 cubic feet.<BR>\n" );
document.write( "similarly, a length of 5 and a width of 2.5 and a height of 4 will get you a tank with a volume of 50 cubic feet.<BR>\n" );
document.write( "these will not, however, fit the requirements of the equation which was that a square had to be cut out of the ends of the rectangle that were all the same size, and the rectangle had to be 12 by 6 to start with.<BR>\n" );
document.write( "please also not that the solution is not an integer, nor is it a rational number as far as i can tell.<BR>\n" );
document.write( "i checked the rational zero theorem.<BR>\n" );
document.write( "from that, the possible answers that could be are:<BR>\n" );
document.write( "50 = 1 * 50 or 2 * 25 or 5 * 10 = p<BR>\n" );
document.write( "4 = 1 * 4 or 2 * 2 = p<BR>\n" );
document.write( "p/q = 50 or 50/4 or 50/2 or 25 or 25/4 or 25/2 or 5 of 5/4 or 5/2 or 10 or 10/4 or 10/2<BR>\n" );
document.write( "these can be simplified to 50 or 25 or 5 or 10 or 12.5 or 6.25 or 2.5 or 1.25 <BR>\n" );
document.write( "none of these turned out to be rational factors of the equation.<BR>\n" );
document.write( "if you want to learn how to find the rational factors of an equation, then this reference might be helpful:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.sparknotes.com/math/algebra2/polynomials/section4.rhtml\" target = \"_blank\">http://www.sparknotes.com/math/algebra2/polynomials<BR>\n" );
document.write( "/section4.rhtml</a><BR>\n" );
document.write( "i have solved problems like this before and they have had solutions.  this particular equation, however, doesn't.<BR>\n" );
document.write( "not as far as i can tell.<BR>\n" );
document.write( "the graph pretty much tells the whole story.<BR>\n" );
document.write( "if the equation is correct (and i believe that it is), the graph shows you there is only one solution to this problem and that solution is not a viable one because it leads to negative lengths and widths.<BR>\n" );
document.write( "you can still get a positive volume, because a minus times a minus equals a plus, but that doesn't help.  the individual lengths and widths both have to be positive.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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