document.write( "Question 582444: If an open box is made from a tin sheet 7 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. (Round your answers to two decimal places.)
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Algebra.Com's Answer #372073 by Edwin McCravy(20056)\"\" \"About 
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If an open box is made from a tin sheet 7 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. (Round your answers to two decimal places.)
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document.write( "We draw the little squares with sides x to cut out of each corner, \r\n" );
document.write( "which will divide each side of the 7\"x7\" square into parts x, 7-2x, \r\n" );
document.write( "and x inches as shown\r\n" );
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document.write( "We cut out the squares and draw in the base of the box (in green):\r\n" );
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document.write( "Now once those sides are bent upward the box will have the\r\n" );
document.write( "dimensions: (7-2x) by (7-2x) by x.  So the volume, V,\r\n" );
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document.write( "   Volume = (length)·(width)·*height)\r\n" );
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document.write( "Let y = volume\r\n" );
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document.write( "    y = (7 - x)·(7 - x)·x\r\n" );
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document.write( "Multiply the first two factor out by FOIL:\r\n" );
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document.write( "    y = (49 - 28x + 4x²)·x\r\n" );
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document.write( "Then multiply the expreeion in parentheses by x:\r\n" );
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document.write( "    y = 49x - 28x² + 4x³\r\n" );
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document.write( "Let's draw the graph:\r\n" );
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document.write( "\"graph%28300%2C300%2C-1%2C4%2C-10%2C30%2C49x-28x%5E2%2B4x%5E3%29\"\r\n" );
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document.write( "We want to find the value of x corresponding to that highest\r\n" );
document.write( "peak point, which is where a tangent line is horizontal and\r\n" );
document.write( "therefore has slope 0, as you see by the green horizontal\r\n" );
document.write( "tangent line drawn below:\r\n" );
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document.write( "The derivative of a function is a formula for the slope of a tangent line \r\n" );
document.write( "drawn any any point (x,y) on the graph of that function.  We want to set \r\n" );
document.write( "that slope = 0.  Therefore we find the derivative and set it = 0 and solve \r\n" );
document.write( "for x: \r\n" );
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document.write( "We find the derivative \"dy%2F%28dx%29\":\r\n" );
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document.write( "   \"dy%2F%28dx%29\" = 49 - 56x + 12x²\r\n" );
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document.write( "We set that equal to 0:\r\n" );
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document.write( "   49 - 56x + 12x² = 0\r\n" );
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document.write( "Get it in descending order\r\n" );
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document.write( "   12x² - 56x + 49 = 0\r\n" );
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document.write( "That can be factored:\r\n" );
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document.write( "  (2x - 7)(6x - 7) = 0\r\n" );
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document.write( "Using the zero-factor property:\r\n" );
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document.write( "  2x - 7 = 0;  6x - 7 = 0\r\n" );
document.write( "      2x = 7;      6x = 7\r\n" );
document.write( "       x = \"7%2F2\";       x = \"7%2F6\"\r\n" );
document.write( "       x = 3.5;     x = 1.17 (rounded to two decimal places.\r\n" );
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document.write( "[Notice that the significance of the answer 3.5 is that if\r\n" );
document.write( "we cut 3.5 inch squares out of the original square, there\r\n" );
document.write( "will be nothing left!!! And so the volume is 0 then!  That\r\n" );
document.write( "is the minimum volume, 0, when we cut all the tin away!  \r\n" );
document.write( "Notice on the graph that the curve is tangent to the x-axis \r\n" );
document.write( "at x=3.6 and reaches a minimum value there.] \r\n" );
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document.write( "So the maximum volume is found where x = \"7%2F6\" or 1.17 inches,\r\n" );
document.write( "rounded to two decimal places.  \r\n" );
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document.write( "Edwin
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