document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1027206>Question 1027206</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a wire 10 cm long is cut into two pieces, one of length x, while the other is of length 10-x. each piece is bent into a square.<BR>\n" );
document.write( "a. find a function that models the total area enclosed by the two squares.<BR>\n" );
document.write( "b. find the value of x that gives the minimum total area enclosed by the two squares.<BR>\n" );
document.write( "c. find the minimum total area. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #642454 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=642454\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the area of a square = side * side = side squared.<BR>\n" );
document.write( "s = side, a = area, formula is a = s^2.\r<BR>\n" );
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document.write( "you cut the wire into 2 pieces.<BR>\n" );
document.write( "one piece has a length of x.<BR>\n" );
document.write( "the other piece has a length of 10-x.\r<BR>\n" );
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document.write( "these lengths form the perimeter of the square.\r<BR>\n" );
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document.write( "there are 4 sides to the perimeter of a square.<BR>\n" );
document.write( "x = perimeter, s = side, formula is x = 4s.<BR>\n" );
document.write( "solve for s to get s = x/4.\r<BR>\n" );
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document.write( "the side of the first square is equal to x/4.\r<BR>\n" );
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document.write( "the side of the second square is equal to (10-x)/4.\r<BR>\n" );
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document.write( "the area of each square is equal to s^2.\r<BR>\n" );
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document.write( "the area of the first square is equal to (x/4)^2.\r<BR>\n" );
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document.write( "the area of the second square is equal to ((10-x)/4)^2\r<BR>\n" );
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document.write( "the total area of the two squares is equal to (x/4)^2 + ((10-x)/4)^2.\r<BR>\n" );
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document.write( "simplify each term in this expression to get:\r<BR>\n" );
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document.write( "(x/4)^2 = x^2/16.\r<BR>\n" );
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document.write( "((10-x)/4)^2 = (10-x)^2/16 = (100- 20x + x^2)/16\r<BR>\n" );
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document.write( "the expression become:\r<BR>\n" );
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document.write( "x^2/16 + (100 - 20x + x^2/16)\r<BR>\n" );
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document.write( "factor out the common term of 1/16 to get:\r<BR>\n" );
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document.write( "(1/16) * (x^2 + 100 - 20x + x^2)\r<BR>\n" );
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document.write( "combine like terms and reorder the terms in descending order of degree to get:\r<BR>\n" );
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document.write( "(1/16) * (2x^2 - 20x + 100)\r<BR>\n" );
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document.write( "factor out the common term of 2 to get:\r<BR>\n" );
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document.write( "(2/16) * (x^2 - 10x + 50)\r<BR>\n" );
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document.write( "this expression represents the area of the two squares.\r<BR>\n" );
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document.write( "let y be equal to the area of the 2 squares, and this expression becomes the equation of:\r<BR>\n" );
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document.write( "y = (1/8) * (x^2 - 10x + 50\r<BR>\n" );
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document.write( "this is a quadratic equation that you can factor by setting it equal to 0.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "(1/8) * (x^2 - 10x + 50) = 0\r<BR>\n" );
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document.write( "you can solve for the roots of this equation, but you don't have to.\r<BR>\n" );
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document.write( "you want to find the minimum point of the quadratic.\r<BR>\n" );
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document.write( "the minimum point of the quadratic is found by using the formula of x = -b/2a.\r<BR>\n" );
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document.write( "in this equation, .....\r<BR>\n" );
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document.write( "a = 1<BR>\n" );
document.write( "b = -10<BR>\n" );
document.write( "c = 50\r<BR>\n" );
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document.write( "x = -b/2a = 10/2 = 5.\r<BR>\n" );
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document.write( "the quadratic equation is at a minimum when x = 5.\r<BR>\n" );
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document.write( "the value of y when x is equal to 5 is found by replacing x in the original equation.\r<BR>\n" );
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document.write( "that original equation is equal to y = (1/8) * (x^2 - 10x + 50) after we factored out the gcf.\r<BR>\n" );
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document.write( "the value of y is therefore equal to (1/8) * (5^2 - 10*5 + 50) which is equal to (1/8) * (25 - 50 + 50) which is equal to (1/8) * 25 which is equal to 3.125.\r<BR>\n" );
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document.write( "the minimum point of the quadratic equation is (5,3.125).\r<BR>\n" );
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document.write( "the value of y is 3.125 when the value of x is 5.\r<BR>\n" );
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document.write( "the minimum area is equal to 3.125 square centimeters when the value of x is equal to 5 centimeters.\r<BR>\n" );
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document.write( "you cut a pice 5 centimeters long and the remaining piece is (10-5) = 5 centimeters long.\r<BR>\n" );
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document.write( "those 2 pieces form the 2 squares where the sum of the areas is smallest.\r<BR>\n" );
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document.write( "the graph of the equation of the area of the two squares is shown below.\r<BR>\n" );
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document.write( "the first equation shown is the original equation.\r<BR>\n" );
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document.write( "the second equation shown is the simplified version of the original equation.\r<BR>\n" );
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document.write( "both those equations form the same graph, as they should, because they are equivalent equations.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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