document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=838180>Question 838180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the second rectangular prism has the following characteristics:<BR>\n" );
document.write( "-it has a volume of 72 cubic inches<BR>\n" );
document.write( "-the sume of the three lengths is 13<BR>\n" );
document.write( "-the width is half of the height<BR>\n" );
document.write( "-the length is one more than the width<BR>\n" );
document.write( "-What are the dimensions of the rectangular prism?\r<BR>\n" );
document.write( "\n" );
document.write( "our email address is:  robkimkc@yahoo.com </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #505062 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=505062\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A rectangular prism has rectangular faces.<BR>\n" );
document.write( "It looks like a cereal box.<BR>\n" );
document.write( "At each corner, 3 edges meet.<BR>\n" );
document.write( "You measure the length of those 3 edges in the units of your choice (maybe inches).<BR>\n" );
document.write( "You call one of those measurement length: you call a second one width, and you call the remaining measurement height.<BR>\n" );
document.write( "The volume of the prism is the product of all three edge length measurements.<BR>\n" );
document.write( "If you measure the edges' lengths in inches, the volume calculated in in cubic inches.<BR>\n" );
document.write( " <BR>\n" );
document.write( "In this problem, the volume is given in cubic inches, so we assume the edge lengths were measured in inches.<BR>\n" );
document.write( "Assume that those measurements are whole numbers (of inches).<BR>\n" );
document.write( "(The problem does not say so, but the teacher assumes it is so, and you have to guess, understand and agree that it is so. Besides, it becomes an advanced algebra problem if we do not assume the measurements are whole numbers of inches).<BR>\n" );
document.write( "What 3 numbers would yield 72 when multiplied together?<BR>\n" );
document.write( "Maybe the teacher talked about prime numbers, and finding prime factor of a number.<BR>\n" );
document.write( "If not, the teacher must have been explaining about factors, and divisibility. In that case, I would not add to the confusion by mentioning prime numbers.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH PRIME NUMBERS:<BR>\n" );
document.write( "Prime numbers are 2, 3, 5, 7, 11, and infinite more numbers that are not multiples of other numbers (except 1 and themselves).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=8%2A9 ALIGN=MIDDLE  ALT=\"72=8%2A9\"   DISABLED_style= \"max-width:90%; height:auto;\" > and writing the 9 and 8 as products of prime numbers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=2%2A2%2A2%2A3%2A3 ALIGN=MIDDLE  ALT=\"72=2%2A2%2A2%2A3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We could also add extra factors to have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=1%2A1%2A1%2A2%2A2%2A2%2A3%2A3 ALIGN=MIDDLE  ALT=\"72=1%2A1%2A1%2A2%2A2%2A2%2A3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but the teacher did not think of that, so do not think of it either.<BR>\n" );
document.write( "There are 5 factors in that product, and we only need 3, so we group them into 3 factors, as in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=%282%2A2%2A2%29%2A3%2A3=8%2A3%2A3 ALIGN=MIDDLE  ALT=\"72=%282%2A2%2A2%29%2A3%2A3=8%2A3%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=2%2A%282%2A2%29%2A%283%2A3%29=2%2A4%2A9 ALIGN=MIDDLE  ALT=\"72=2%2A%282%2A2%29%2A%283%2A3%29=2%2A4%2A9\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=%282%2A2%29%2A%282%2A3%29%2A3=4%2A6%2A3 ALIGN=MIDDLE  ALT=\"72=%282%2A2%29%2A%282%2A3%29%2A3=4%2A6%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or .....<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOT NEEDING TO MENTION PRIMES:<BR>\n" );
document.write( "We need three factors that multiplied together give us 72,<BR>\n" );
document.write( "and that also satisfy the other two statements:<BR>\n" );
document.write( "-the sum of the three lengths is 13<BR>\n" );
document.write( "-the width is half of the height.<BR>\n" );
document.write( " <BR>\n" );
document.write( "What factors of 72 do we know?<BR>\n" );
document.write( "How do we find them?<BR>\n" );
document.write( "We try numbers and divide. If a number goes evenly into 72, it is a factor.<BR>\n" );
document.write( "We start from the smallest numbers and go up.<BR>\n" );
document.write( "We may think of 1, because all numbers are divisible by 1.<BR>\n" );
document.write( "(The teacher may not have thought of 1, but should not be deeply offended if we do).<BR>\n" );
document.write( "We know 72 is divisible by 2, and by 3, and by 4.<BR>\n" );
document.write( "It is not divisible by 5, or by 7,<BR>\n" );
document.write( "but it is divisible by 6, by 8 and by 9.<BR>\n" );
document.write( "We do not need to keep trying larger number, because factors come in matched pairs.<BR>\n" );
document.write( "When we divide a number by one of its factors, we find another factor, so each smaller factor we find comes with a matching larger factor.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=72=8%2A9 ALIGN=MIDDLE  ALT=\"72=8%2A9\"   DISABLED_style= \"max-width:90%; height:auto;\" > implies 72 ÷ 8 = 9,<BR>\n" );
document.write( "so when we find 8, we find 9.<BR>\n" );
document.write( "Similarly, from 1 we find 72 ÷ 1 = 72,<BR>\n" );
document.write( "from 2 we find 72 ÷ 2 = 36,<BR>\n" );
document.write( "from 3 we find 72 ÷ 3 = 24,<BR>\n" );
document.write( "from 4 we find 72 ÷ 4 = 18, and<BR>\n" );
document.write( "from 6 we find 72 ÷ 6 = 12.<BR>\n" );
document.write( "Once we find 8 and it brings 9 as a match, we do not need to go further.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If one of the factors (width) is going to be half of another factor (height), <BR>\n" );
document.write( "the possibilities we have to try are:<BR>\n" );
document.write( "1 for the width and 2 for the height, or<BR>\n" );
document.write( "2 for the width and 4 for the height, or<BR>\n" );
document.write( "3 for the width and 6 for the height, or<BR>\n" );
document.write( "6 for the width and 12 for the height.<BR>\n" );
document.write( "Nothing else works, because we cannot have a width of 4 inches and a height of 8 inches.<BR>\n" );
document.write( "(Those two factors, 4 and 8, do not fit together into 72).<BR>\n" );
document.write( " <BR>\n" );
document.write( "We are going to try and see if one of those combinations work.<BR>\n" );
document.write( "Maybe the teacher told them to organize their work in a table with four columns for width, height, length, and SUM of all 3.<BR>\n" );
document.write( "It's to hard to draw such a table here.<BR>\n" );
document.write( "If the measurements in inches are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=width=2 ALIGN=MIDDLE  ALT=\"width=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=height=4 ALIGN=MIDDLE  ALT=\"height=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=width%2Aheight=2%2A4=8 ALIGN=MIDDLE  ALT=\"width%2Aheight=2%2A4=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the third factor (the length) has to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SUM=2%2B4%2B9=15 ALIGN=MIDDLE  ALT=\"SUM=2%2B4%2B9=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so that combination does not work.<BR>\n" );
document.write( " <BR>\n" );
document.write( "If the measurements in inches are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=width=3 ALIGN=MIDDLE  ALT=\"width=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=height=6 ALIGN=MIDDLE  ALT=\"height=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=width%2Aheight=3%2A6=18 ALIGN=MIDDLE  ALT=\"width%2Aheight=3%2A6=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the third factor (the length) has to be 72 ÷ 18 = 4.<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=length=4 ALIGN=MIDDLE  ALT=\"length=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SUM=3%2B6%2B4=13 ALIGN=MIDDLE  ALT=\"SUM=3%2B6%2B4=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so we found a solution.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The other possibilities do not work:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28width=1%2Cheight=2%29 ALIGN=MIDDLE  ALT=\"system%28width=1%2Cheight=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=length=36 ALIGN=MIDDLE  ALT=\"length=36\"   DISABLED_style= \"max-width:90%; height:auto;\" > No good. (SUM=1+2+36=39)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28width=6%2Cheight=12%29 ALIGN=MIDDLE  ALT=\"system%28width=6%2Cheight=12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=length=1 ALIGN=MIDDLE  ALT=\"length=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > No good. (Length should be larger than than width, and they do not add up to 13 either).  <BR>\n" );
document.write( " <BR>\n" );
document.write( "The dimensions of the rectangular prism are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28width=3inches%2Cheight=6inches%2Clength=4inches%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28width=3inches%2Cheight=6inches%2Clength=4inches%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );