document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=60127>Question 60127</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780073268620/algebrahomeworkh>Elementry and Intermediate Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> A cubic shipping container had a volume of v^3 cubic meters. The height of the container was decreased by a whole number of meters and the width was increased by a whole number of meters so that the volume of the container is now v^3 + 2v^2 – 3v. Find out the following:\r<BR>\n" );
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document.write( "a.	By how many meters the height was decreased?\r<BR>\n" );
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document.write( "b.	By how many meters the width was increased?  <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #41320 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=41320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">A cubic shipping container had a volume of v^3 cubic meters. The height of the\r\n" );
document.write( "container was decreased by a whole number of meters and the width was increased\r\n" );
document.write( "by a whole number of meters so that the volume of the container is now \r\n" );
document.write( "v^3 + 2v^2 – 3v. Find out the following:\r\n" );
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document.write( "a.	By how many meters the height was decreased?\r\n" );
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document.write( "b.	By how many meters the width was increased?  \r\n" );
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document.write( "volume of container originally = v³ = \r\n" );
document.write( " original length × original width × original height\r\n" );
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document.write( "Let l = original length\r\n" );
document.write( "Let w = original width\r\n" );
document.write( "Let h = original height\r\n" );
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document.write( "Let a = the whole number of meters the height was decreased\r\n" );
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document.write( "Let b = the whole number of meters the width was increased\r\n" );
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document.write( "Then\r\n" );
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document.write( "new length = original length = l (length was not changed)\r\n" );
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document.write( "new width = (original width + b) = w + b\r\n" );
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document.write( "new height = (original height - a) = h - a\r\n" );
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document.write( "new volume = v³ + 2v² - 3v  \r\n" );
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document.write( "new volume = new length × new width × new height\r\n" );
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document.write( "v³ + 2v² - 3v = l(w + b)(h - a)\r\n" );
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document.write( "v³ + 2v² - 3v = l(w + b)(h - a)\r\n" );
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document.write( "v³ + 2v² - 3v = l(wh - aw + bh - ab)\r\n" );
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document.write( "v³ + 2v² - 3v = lwh - law + lbh - lab\r\n" );
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document.write( "Since v³ = lwh we subtract v³ from left side and\r\n" );
document.write( "subtract lwh from right side\r\n" );
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document.write( "     2v² - 3v = -law + lbh - lab\r\n" );
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document.write( "Rearranging terms on righ so that positive term comes first\r\n" );
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document.write( "     2v² - 3v = lbh - law - lab \r\n" );
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document.write( "Factor out v on left; factor out l on right\r\n" );
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document.write( "    v(2v - 3) = l(bh - aw - ab)\r\n" );
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document.write( "As it turns out there are infinitely many solutions to that.\r\n" );
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document.write( "Here are a few:\r\n" );
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document.write( "	v = 2, l = 1, w = 1, b = 1, a = 3, h = 8\r\n" );
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document.write( "original volume = original length × original width × original height\r\n" );
document.write( "original volume = lwh = 1(1)(8) = 8\r\n" );
document.write( "original volume = v³ = 2³ = 8 \r\n" );
document.write( "That checks!\r\n" );
document.write( "new length = original length = 1 (length was not changed)\r\n" );
document.write( "new width = (original width + b) = 1 + 1 = 2\r\n" );
document.write( "new height = (original height - a) = 8 - 3 = 5\r\n" );
document.write( "new volume = v³ + 2v² - 3v = 2³ + 2(2)² - 3(2) = 10  \r\n" );
document.write( "new volume = new length × new width × new height\r\n" );
document.write( "new volume = 1(2)(5) = 10\r\n" );
document.write( "That checks!  Answer: height decreased by a = 3,\r\n" );
document.write( "width increased by b = 1\r\n" );
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document.write( "        v = 5, l = 5, w = 5, b = 11, a = 3, h = 5\r\n" );
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document.write( "original volume = original length × original width × original height\r\n" );
document.write( "original volume = lwh = 5(5)(5) = 125\r\n" );
document.write( "original volume = v³ = 5³ = 125\r\n" );
document.write( "That checks! \r\n" );
document.write( "new length = original length = 5 (length was not changed)\r\n" );
document.write( "new width = (original width + b) = 5 + 11 = 16\r\n" );
document.write( "new height = (original height - a) = 5 - 3 = 2\r\n" );
document.write( "new volume = v³ + 2v² - 3v = 5³ + 2(5)² - 3(5) = 160  \r\n" );
document.write( "new volume = new length × new width × new height\r\n" );
document.write( "new volume = 5(16)(2) = 160\r\n" );
document.write( "That checks!  Answer: height decreased by a = 3,\r\n" );
document.write( "width increased by b = 11\r\n" );
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document.write( "        v = 6, l = 9, w = 4, b = 6, a = 3, h = 6\r\n" );
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document.write( "original volume = original length × original width × original height\r\n" );
document.write( "original volume = lwh = 9(4)(6) = 216\r\n" );
document.write( "original volume = v³ = 6³ = 216\r\n" );
document.write( "That checks!\r\n" );
document.write( "new length = original length = 9 (length was not changed)\r\n" );
document.write( "new width = (original width + b) = 4 + 6 = 10\r\n" );
document.write( "new height = (original height - a) = 6 - 3 = 3\r\n" );
document.write( "new volume = v³ + 2v² - 3v = 6³ + 2(6)² - 3(6) = 270  \r\n" );
document.write( "new volume = new length × new width × new height\r\n" );
document.write( "new volume = 9(10)(3) = 270\r\n" );
document.write( "That checks!  Answer: height decreased by a = 3,\r\n" );
document.write( "width increased by b = 6\r\n" );
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document.write( "        v = 7, l = 7, w = 7, b = 5, a = 2, h = 7\r\n" );
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document.write( "original volume = original length × original width × original height\r\n" );
document.write( "original volume = lwh = 7(7)(7) = 343\r\n" );
document.write( "original volume = v³ = 7³ = 343\r\n" );
document.write( "That checks!\r\n" );
document.write( "new length = original length = 7 (length was not changed)\r\n" );
document.write( "new width = (original width + b) = 7 + 5 = 12\r\n" );
document.write( "new height = (original height - a) = 7 - 2 = 5\r\n" );
document.write( "new volume = v³ + 2v² - 3v = 7³ + 2(7)² - 3(7) = 420  \r\n" );
document.write( "new volume = new length × new width × new height\r\n" );
document.write( "new volume = 7(12)(5) = 420\r\n" );
document.write( "That checks!  Answer: height decreased by a = 2,\r\n" );
document.write( "width increased by b = 5\r\n" );
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document.write( "        v = 12, l = 12, w = 12, b = 3, a = 1, h = 12\r\n" );
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document.write( "original volume = original length × original width × original height\r\n" );
document.write( "original volume = lwh = 12(12)(12) = 1728\r\n" );
document.write( "original volume = v³ = 12³ = 1728\r\n" );
document.write( "That checks!\r\n" );
document.write( "new length = original length = 12 (length was not changed)\r\n" );
document.write( "new width = (original width + b) = 12 + 3 = 15\r\n" );
document.write( "new height = (original height - a) = 12 - 1 = 11\r\n" );
document.write( "new volume = v³ + 2v² - 3v = 12³ + 2(12)² - 3(12) = 1980  \r\n" );
document.write( "new volume = new length × new width × new height\r\n" );
document.write( "new volume = 12(15)(11) = 1980\r\n" );
document.write( "That checks!  Answer: height decreased by a = 1,\r\n" );
document.write( "width increased by b = 3\r\n" );
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document.write( "        v = 15, l = 15, w = 15, b = 13, a = 6, h = 15\r\n" );
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document.write( "original volume = original length × original width × original height\r\n" );
document.write( "original volume = lwh = 15(15)(15) = 3375\r\n" );
document.write( "original volume = v³ = 15³ = 3375\r\n" );
document.write( "That checks!\r\n" );
document.write( "new length = original length = 15 (length was not changed)\r\n" );
document.write( "new width = (original width + b) = 15 + 13 = 28\r\n" );
document.write( "new height = (original height - a) = 15 - 6 = 9\r\n" );
document.write( "new volume = v³ + 2v² - 3v = 15³ + 2(15)² - 3(15) = 3780  \r\n" );
document.write( "new volume = new length × new width × new height\r\n" );
document.write( "new volume = 15(28)(9) = 3780\r\n" );
document.write( "That checks!  Answer: height decreased by a = 6,\r\n" );
document.write( "width increased by b = 13\r\n" );
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document.write( "There are billions and billions more solutions!!! \r\n" );
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document.write( "Edwin</PRE></font></b><BR>\n" );
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