document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=410522>Question 410522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Minimizing Area) A 24in. piece of string is cut into two pieces. One piece is used to form a circle while the other is used to form a square. How should the string be cut so that the sum of the areas is a minimum? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #288658 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=288658\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The smallest area is equal to 1013.53155 as shown in the graph below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-24%2C24%2C-100%2C2000%2C7.141592654%2Ax%5E2+-+192%2Ax+%2B+2304%2C1013.53155%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the full picture of this graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C800%2C-24%2C24%2C-1000%2C15000%2C7.141592654%2Ax%5E2+-+192%2Ax+%2B+2304%2C1013.53155%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the solution was derived as follows:\r<BR>\n" );
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document.write( "let x = length of string that forms the square.<BR>\n" );
document.write( "let 24-x = length of string that forms the circle.\r<BR>\n" );
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document.write( "circumference of a square is 4*s where s is equal to 1 side of the square.\r<BR>\n" );
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document.write( "since the length of the string forms the circumference of the square, we get:\r<BR>\n" );
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document.write( "4*s = x which allows us to derive:\r<BR>\n" );
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document.write( "s = (x/4)\r<BR>\n" );
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document.write( "area of a square is equal to s^2 which is therefore equal to (x/4)^2 which is therefore equal to x^2/16\r<BR>\n" );
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document.write( "area of the square is equal to x^2/16\r<BR>\n" );
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document.write( "the circumference of a circle is equal to 2*pi*r\r<BR>\n" );
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document.write( "since the length  of the remaining string is equal to the circumference of the circle, then we get:\r<BR>\n" );
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document.write( "2*pi*r = 24-x\r<BR>\n" );
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document.write( "solve for r to get:\r<BR>\n" );
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document.write( "r = (24-x) / (2*pi)\r<BR>\n" );
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document.write( "the area of a circle is equal to pi * r^2\r<BR>\n" );
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document.write( "substitute to get:\r<BR>\n" );
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document.write( "area of a circle is equal to pi * ((24-x)^2/(2*pi))^2\r<BR>\n" );
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document.write( "simplify this to get:\r<BR>\n" );
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document.write( "area of a circle is equal to pi * ((24-x)^2/(4*pi^2)\r<BR>\n" );
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document.write( "simplify this further to get:\r<BR>\n" );
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document.write( "area of a circle is equal to (24-x)^2 / (4*pi)\r<BR>\n" );
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document.write( "simplify this further to get:\r<BR>\n" );
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document.write( "area of a circle is equal to (x^2 - 48x + 576) / (4*pi)\r<BR>\n" );
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document.write( "let the sum of the area of the square and the circle equal to y.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "y = x^2/16 + ((x^2 - 48x + 576) / (4*pi))\r<BR>\n" );
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document.write( "simplify this by finding a common denominator to get:\r<BR>\n" );
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document.write( "y = (4*pi*x^2 + 16 * (x^2 - 48*x + 576)) / (4*16*pi)\r<BR>\n" );
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document.write( "simplify this further to get:\r<BR>\n" );
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document.write( "y = (4*pi*x^2 + 16*x^2 - 768*x + 9216) / (64*pi)\r<BR>\n" );
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document.write( "set y = 0 to get:\r<BR>\n" );
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document.write( "(4*pi*x^2 + 16*x^2 - 768*x + 9216) / (64*pi) = 0\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "(4*pi + 16)*x^2 - 768*x + 9216) / 64*pi) = 0\r<BR>\n" );
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document.write( "this is your quadratic equation in standard form.\r<BR>\n" );
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document.write( "multiply both sides of this equation by (64*pi) to get:\r<BR>\n" );
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document.write( "(4*pi + 16)*x^2 - 768*x + 9216) = 0\r<BR>\n" );
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document.write( "divide both sides of this equation by 4 to get:\r<BR>\n" );
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document.write( "(pi+4)*x^2 - 192*x + 2304 = 0\r<BR>\n" );
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document.write( "replace pi with it's constant value of 3.141592654 and combine like terms to get:\r<BR>\n" );
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document.write( "7.141592654*x^2 - 192*x + 2304 = 0\r<BR>\n" );
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document.write( "in this quadratic equation:\r<BR>\n" );
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document.write( "a = 7.141592654<BR>\n" );
document.write( "b = -192<BR>\n" );
document.write( "c = 2304\r<BR>\n" );
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document.write( "max/min point is found by using the formula of x = -b/2a\r<BR>\n" );
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document.write( "x = -b/2a becomes:\r<BR>\n" );
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document.write( "x = 192/14.28318531 which becomes:\r<BR>\n" );
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document.write( "x = 13.44237968\r<BR>\n" );
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document.write( "substitute for x in the equation of:\r<BR>\n" );
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document.write( "(4*pi + 16)*x^2 - 768*x + 9216) to get:\r<BR>\n" );
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document.write( "when x = 13.44237968, y equals 1013.53155\r<BR>\n" );
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document.write( "that's the minimum point of your quadratic equation which is modelling the sum of the areas of the square and the circle.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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