document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1098180>Question 1098180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the length of the edge of a cube if after a slice 1cm thick is cut from one side, the volume remaining 294 cubic cm? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #712591 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=712591\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> THE CARPENTER (OR THE CONFIDENT FIFTH-GRADER) SOLUTION:<BR>\n" );
document.write( "I know how to calculate the volume of any shoe-box-shaped item.<BR>\n" );
document.write( "You just multiply width, length and height.<BR>\n" );
document.write( "For a cube, width length and height are all the same edge length.<BR>\n" );
document.write( "If the cube edge length was 10 cm,<BR>\n" );
document.write( "the initial cube volume would be<BR>\n" );
document.write( "(10 cm) X (10 cm) X (10 cm) = 1000 cubic cm.<BR>\n" );
document.write( "The volume of the slice cut off would be<BR>\n" );
document.write( "(10 cm) X (10 cm) X (1 cm) = 100 cubic cm.<BR>\n" );
document.write( "So, the final volume would be 900 cubic cm.<BR>\n" );
document.write( "That is more than the 249 cm in the question.<BR>\n" );
document.write( " <BR>\n" );
document.write( "I also know that the larger the original cube,<BR>\n" );
document.write( "the larger the final volume,<BR>\n" );
document.write( "and the larger the final volume required,<BR>\n" );
document.write( "the larger the original cube needed.<BR>\n" );
document.write( " <BR>\n" );
document.write( "What I know tells mne that the original cube edge must be less than 10 cm long.<BR>\n" );
document.write( "It also tells me that there is only one answer.<BR>\n" );
document.write( "If I calculate using increasing whole number lengths,<BR>\n" );
document.write( "the calculated volumes will keep increasing,<BR>\n" );
document.write( "so I will get to the answer and 249 cubic cm at some point,<BR>\n" );
document.write( "or the answer was not a whole number length, and I will go past the answer.\r<BR>\n" );
document.write( "\n" );
document.write( "I can calculate what the volume would be for a few edge lengths to see if I get 249 cubic cm as the answer.\r<BR>\n" );
document.write( "\n" );
document.write( "If that happens, I will have the answer.<BR>\n" );
document.write( "If not,<BR>\n" );
document.write( "I will get less than 249 cubic cm for some whole number edge length,<BR>\n" );
document.write( "but more than 249 cubic cm for the next whole number edge length,<BR>\n" );
document.write( "and I will know that the answer is somewhere in between.<BR>\n" );
document.write( " <BR>\n" );
document.write( "It is only common sense that the final volume is<BR>\n" );
document.write( "more than the volume of a cube with edges 1 cm smaller.<BR>\n" );
document.write( "I can easily calculate that a cube with edge length 5 cm has a volume of 125 cubic cm,<BR>\n" );
document.write( "so the original cube's edge must be larger than 5 cm.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%284%2C5%2C%0D%0Aedge%2Clength%2C5cm%2C6cm%2Chighlight%287cm%29%2C%0D%0Acube%2C+volume%2C%22125+cu.cm%22%2C%22216+cu.cm%22%2C%22343+cu.cm%22%2C%0D%0Aslice%2C+volume%2C%2225+cu.cm%22%2C%2236+cu.cm%22%2C%2249+cu.cm%22%2C%0D%0Afinal%2Cvolume%2C%22100+cu.cm%22%2C%22180+cu.cm%22%2C%22249+cu.cm%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "THE HIGH-SCHOOLER SOLUTION:<BR>\n" );
document.write( "If<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= length of the edge of the cube, in cm,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >= volume of the cube in cubic cm, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Ax%2A1=x%5E2 ALIGN=MIDDLE  ALT=\"x%2Ax%2A1=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= volume of a slice 1cm thick is cut from one side of the cube.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-x%5E2=294 ALIGN=MIDDLE  ALT=\"x%5E3-x%5E2=294\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the volume remaining, in cubic cm.<BR>\n" );
document.write( " <BR>\n" );
document.write( "All you have to do is solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-x%5E2=294 ALIGN=MIDDLE  ALT=\"x%5E3-x%5E2=294\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-x%5E2-294=0 ALIGN=MIDDLE  ALT=\"x%5E3-x%5E2-294=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3-x%5E2=294 ALIGN=MIDDLE  ALT=\"x%5E3-x%5E2=294\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29x%5E2=294 ALIGN=MIDDLE  ALT=\"%28x-1%29x%5E2=294\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It so happens that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=294=2%2A3%2A7%5E2=6%2A7%5E2 ALIGN=MIDDLE  ALT=\"294=2%2A3%2A7%5E2=6%2A7%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Is that the only solution?<BR>\n" );
document.write( "The way the problwm is worded,<BR>\n" );
document.write( "you would think that there is only one solution,<BR>\n" );
document.write( "so answering <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%287cm%29 ALIGN=MIDDLE  ALT=\"highlight%287cm%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and going to the next problem would be a good strategy.<BR>\n" );
document.write( " <BR>\n" );
document.write( "With time and willingless to spare, you could dig deeper<BR>\n" );
document.write( "using whatever tools you have.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using a graphing calculator, you could graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3-x%5E2-294 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3-x%5E2-294\"   DISABLED_style= \"max-width:90%; height:auto;\" > as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C300%2C-7%2C9%2C-450%2C50%2Cx%5E3-x%5E2-294%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C300%2C-7%2C9%2C-450%2C50%2Cx%5E3-x%5E2-294%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Using calculus:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3-x%5E2-294 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3-x%5E2-294\"   DISABLED_style= \"max-width:90%; height:auto;\" > could have 1 or 3 real zeros.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=3x%5E2-2x=x%283x-2%29 ALIGN=MIDDLE  ALT=\"df%2Fdx=3x%5E2-2x=x%283x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has zeros at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%2F3 ALIGN=MIDDLE  ALT=\"x=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "representing respectively a local maximum and a local minimum for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=-294 ALIGN=MIDDLE  ALT=\"f%280%29=-294\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > at its local maximum,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+increases+for+%7B%7B%7Bx%3C0 ALIGN=MIDDLE  ALT=\"f%28x%29+increases+for+%7B%7B%7Bx%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to local maximum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%280%29=-294 ALIGN=MIDDLE  ALT=\"f%280%29=-294\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "decreases at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3Cx%3C2%2F3 ALIGN=MIDDLE  ALT=\"0%3Cx%3C2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to a local minimum at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2%2F3 ALIGN=MIDDLE  ALT=\"x=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <BR>\n" );
document.write( "and then increases for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E2%2F3 ALIGN=MIDDLE  ALT=\"x%3E2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, there can be only one real zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "it happens for some <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E2%2F3 ALIGN=MIDDLE  ALT=\"x%3E2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and as we already found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7 ALIGN=MIDDLE  ALT=\"x=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a zero, we know know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only real zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=x%5E3-x%5E2-294=0 ALIGN=MIDDLE  ALT=\"f%28x%29=x%5E3-x%5E2-294=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );