document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=973731>Question 973731</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Quite long: I have the answer already, I would like to make sure I did it correctly, thank you!\r<BR>\n" );
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document.write( "Ship it here is a comoany that specializes in shopping freght in medium sized boxes. The company limits the boxes that it will ship to those who's sum of the length, height and width is at least 100 cm but no more then a 1000. Gadgets to go is another comoany that ships their products  in boxes which has integer dimensiones such that their length, width, height are in the rations 4:3:5. The largest they use to ship any product has a volume less then 2 cubic centimeters. Determine the dimensions of both the smallest and largest which can be shipped by gadgets to go and ship it here.\r<BR>\n" );
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document.write( "I basically took 4x+3x+5x=100 and 4x x 5x x 3x\r<BR>\n" );
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document.write( "From the on I used guess and check.\r<BR>\n" );
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document.write( "Here are my answers:\r<BR>\n" );
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document.write( "The largest box has: 128, 96 and 160 as their dimensions <BR>\n" );
document.write( "Smallest: 36,27 and 45\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595812 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=595812\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Guess and check works easily if you can use a computer (preferably with a spreadsheet program),<BR>\n" );
document.write( "or a graphing calculator,<BR>\n" );
document.write( "but it is time consuming otherwise.<BR>\n" );
document.write( "There is an algebra way to get to the solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "You figured out that for the boxes to have dimensions (in centimeters) that are integers, and are in the ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%3A4%3A5 ALIGN=MIDDLE  ALT=\"3%3A4%3A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the dimensions (in centimeters) must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x ALIGN=MIDDLE  ALT=\"4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >= a positive integer.<BR>\n" );
document.write( "That is one of the requirements of the company Gadgets To Go.<BR>\n" );
document.write( "For a box meeting that requirement, the sum of the dimensions (in centimeters) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4x%2B5x=12x ALIGN=MIDDLE  ALT=\"3x%2B4x%2B5x=12x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For that box to meet the requirements of the company Ship It Here, it must be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%3C=12x%3C=1000 ALIGN=MIDDLE  ALT=\"100%3C=12x%3C=1000\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2F12%3C=x%3C=1000%2F12 ALIGN=MIDDLE  ALT=\"100%2F12%3C=x%3C=1000%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2F12 ALIGN=MIDDLE  ALT=\"100%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >=8.3333...  and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1000%2F12 ALIGN=MIDDLE  ALT=\"1000%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >=83.3333... ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be integer, it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%3C=x%3C=83 ALIGN=MIDDLE  ALT=\"9%3C=x%3C=83\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We calculate the volume (in cubic centimeters) of a box measuring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x ALIGN=MIDDLE  ALT=\"4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > cm as <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Volume=%283x%29%2A%284x%29%2A%285x%29=60x%5E3 ALIGN=MIDDLE  ALT=\"Volume=%283x%29%2A%284x%29%2A%285x%29=60x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=9 ALIGN=MIDDLE  ALT=\"x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x=27%2C4x=36%2C5x=45%2CVolume=60%2A9%5E3=60%2A725=43740%29 ALIGN=MIDDLE  ALT=\"system%283x=27%2C4x=36%2C5x=45%2CVolume=60%2A9%5E3=60%2A725=43740%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That agrees with the dimensions you list for the smallest box.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=83 ALIGN=MIDDLE  ALT=\"x=83\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x=249%2C4x=332%2C5x=415%2CVolume=60%2A9%5E3=60%2A725=34307220%29 ALIGN=MIDDLE  ALT=\"system%283x=249%2C4x=332%2C5x=415%2CVolume=60%2A9%5E3=60%2A725=34307220%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Gadgets To Go seems to have a maximum volume limit lower than 34,307,220 cubic centimeters,<BR>\n" );
document.write( "but \"The largest they use to ship any product has a volume less then 2 cubic centimeters\" does not make sense.<BR>\n" );
document.write( "The smallest box Gadgets To Go would ship would measure 3 cm by 4 cm by 5 cm,<BR>\n" );
document.write( "and its volume (in cubic centimeters) would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A4%2A5=60 ALIGN=MIDDLE  ALT=\"3%2A4%2A5=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The volume of largest box Gadgets To Go would ship cannot be \"less then 2 cubic centimeters.\"<BR>\n" );
document.write( " <BR>\n" );
document.write( "If it were less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V ALIGN=MIDDLE  ALT=\"V\"   DISABLED_style= \"max-width:90%; height:auto;\" > cubic centimeters,<BR>\n" );
document.write( "we would require<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Volume=60x%5E3%3C=V ALIGN=MIDDLE  ALT=\"Volume=60x%5E3%3C=V\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%3C=V%2F60 ALIGN=MIDDLE  ALT=\"x%5E3%3C=V%2F60\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=root%283%2CV%2F60%29 ALIGN=MIDDLE  ALT=\"x%3C=root%283%2CV%2F60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If it were that \"The largest they use to ship any product has a volume less than 2 cubic meters\" = 2,000,000 centimeters,<BR>\n" );
document.write( "we would require <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=root%283%2C2000000%2F60%29 ALIGN=MIDDLE  ALT=\"x%3C=root%283%2C2000000%2F60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C2000000%2F60%29=about32.18 ALIGN=MIDDLE  ALT=\"root%283%2C2000000%2F60%29=about32.18\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we would settle for integers such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=32 ALIGN=MIDDLE  ALT=\"x%3C=32\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which meets the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%3C=x%3C=83 ALIGN=MIDDLE  ALT=\"9%3C=x%3C=83\"   DISABLED_style= \"max-width:90%; height:auto;\" > requirement and would result in a largest box with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x=3%2A32=96%2C4x=4%2A32=128%2C5x=5%2A32=160%2CVolume=60%2A32%5E3=60%2A725=1966080%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That agrees with the dimensions you list for the largest box. </SPAN>\n" );
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