document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027670>Question 1027670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube?   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642910 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=642910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> formula for surface area of a cube is 6s^2.\r<BR>\n" );
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document.write( "s represents the length of the side of the cube.\r<BR>\n" );
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document.write( "that means that 6s^2 = 6x^2 - 36x + 54.\r<BR>\n" );
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document.write( "divide both sides of this equation by 6 and you get:\r<BR>\n" );
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document.write( "s^2 = x^2 - 6x + 9\r<BR>\n" );
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document.write( "factor that quadratic equation on the right side of the equation and you get:\r<BR>\n" );
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document.write( "s^2 = (x-3) * (x-3) = (x-3)^2\r<BR>\n" );
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document.write( "take the square root of both sides of the equation to get:\r<BR>\n" );
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document.write( "s = plus or minus (x-3).\r<BR>\n" );
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document.write( "since s can't be negative, then s = (x-3).\r<BR>\n" );
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document.write( "note that the value of x would have to be greater than 3, otherwise the length of the side would be negative.\r<BR>\n" );
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document.write( "the formula for the volume of a cube is s^3.\r<BR>\n" );
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document.write( "that means the volume of the cube must be equal to (x-3)^3.\r<BR>\n" );
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document.write( "since (x-3)^2 = x^2 - 6x + 9, then (x-3)^3 must be equal to:\r<BR>\n" );
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document.write( "(x-3) * (x^2 - 6x + 9) which would then be equal to x^3 - 9x^2 + 27x - 27.\r<BR>\n" );
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document.write( "how do you confirm this is correct?\r<BR>\n" );
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document.write( "assume the side of the cube is equal to 5 units.\r<BR>\n" );
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document.write( "the surface area would be 6s^2 = 6 * 5^2 = 6 * 25 = 150 square units.\r<BR>\n" );
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document.write( "the volume would be s^3 = 5^3 = 125 cubic units.\r<BR>\n" );
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document.write( "we had previously shown that s = x-3.\r<BR>\n" );
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document.write( "this means that x = s + 3.\r<BR>\n" );
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document.write( "if s = 5, this means that x = 8.\r<BR>\n" );
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document.write( "our formula for surface area is 6 * s^2 = 6.\r<BR>\n" );
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document.write( "since s = (x-3), then the surface area = 6 * (x-3)^2 = 6x^2 - 36x + 54.\r<BR>\n" );
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document.write( "replace x with 8 in these formulas and you will see that 6 * (x-3)^2 = 6 * 5^2 = 6 * 25 = 150 square units.\r<BR>\n" );
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document.write( "you will also see that 6x^2 - 36x + 54 = 6 * 8^2 - 36 * 8 + 54 = 384 - 288 + 54 = 150 square units.\r<BR>\n" );
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document.write( "the volume of the cube is equal to s^3.\r<BR>\n" );
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document.write( "when s = 5, this gets you the volume of the cube = 125 cubic units.\r<BR>\n" );
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document.write( "when s = 5, x = 8.\r<BR>\n" );
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document.write( "the volume of the cube = (x-3)^3 = (8-5)^3 = 5^3 = 125 cubic units.\r<BR>\n" );
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document.write( "the volume of the cube is also equal to x^3 - 9x^2 + 27x - 27.\r<BR>\n" );
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document.write( "when x = 8, this becomes (8)^3 - 9*8^2 + 27*8 - 27 which becomes equal to 512- 576 + 216 - 27 which becomes equal to 125 cubic units.\r<BR>\n" );
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document.write( "the formulas look good.\r<BR>\n" );
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document.write( "the expression for the volume of the cube would be (x-3)^3 or x^3 - 9x^2 + 27x - 27, depending on how you want to show it.\r<BR>\n" );
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