I am a freshman in honors algebra 2 and we're working on applying prabolas to real world senarios. The question is \" find the dimensions and maximum area of a rectangle with a perimeter of 48 inches\" all I have now is basic rectangle properties because I spaced off in class. Please help me.
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document.write( " A = L×W where A = area, L = length and W = wifth\r\n" );
document.write( " P = 2L + 2W where P = perimeter\r\n" );
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document.write( "Let the length be x\r\n" );
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document.write( "Since the perimeter is 48, we substitute in\r\n" );
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document.write( " P = 2L + 2W\r\n" );
document.write( " 48 = 2x + 2W\r\n" );
document.write( "Divide every term by 2\r\n" );
document.write( " 24 = x + W\r\n" );
document.write( "Solve for W\r\n" );
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document.write( " 24 - x = W\r\n" );
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document.write( "Let the area be y. Substitute in\r\n" );
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document.write( " A = L×W\r\n" );
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document.write( " y = x(24 - x)\r\n" );
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document.write( "That equation has this graph: \r\n" );
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document.write( "Since y = the area, the area will be a maximum at the\r\n" );
document.write( "peak point, known as \"the vertex\".\r\n" );
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document.write( "There are three ways to find the vertex. I don't know which way\r\n" );
document.write( "your teacher will require you to use.\r\n" );
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document.write( "Method 1 for finding the vertex:\r\n" );
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document.write( "Find the two x-intercepts\r\n" );
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document.write( " y = x(24 - x)\r\n" );
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document.write( " x(24 - x) = 0\r\n" );
document.write( " x=0, 24-x = 0\r\n" );
document.write( " -x = -24\r\n" );
document.write( " x = 24\r\n" );
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document.write( "The x-intercepts are (0,0) and (24,0)\r\n" );
document.write( "The vertex occurs halfway when the value of x is halfway\r\n" );
document.write( "between 0 and 24, or at x=12.\r\n" );
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document.write( "Substituting x=12\r\n" );
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document.write( " y = 12(24-12)\r\n" );
document.write( " y = 12(12)\r\n" );
document.write( " y = 144\r\n" );
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document.write( "So the vertex of that parabola is (12,144)\r\n" );
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document.write( "That means the area y will have a maximium area of 144 square units\r\n" );
document.write( "when the length is x = 12 inches. That is, when the rectangle is a\r\n" );
document.write( "12in × 12in square. \r\n" );
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document.write( "Method 2 for finding the vertex:\r\n" );
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document.write( " y = x(24 - x)\r\n" );
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document.write( "Put the equation in the vertex form:\r\n" );
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document.write( " y = a(x-h)² + k\r\n" );
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document.write( " y = x(24 - x)\r\n" );
document.write( " y = 24x - x²\r\n" );
document.write( " y = -x² + 24x\r\n" );
document.write( "Factor out the coefficient of x², which is -1\r\n" );
document.write( " y = -1[x² - 24x ]\r\n" );
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document.write( "Complete the square by multiplying the coefficient of x by 
\r\n" );
document.write( "ans squaring: -24() = -12, Squaring (-12)² = 144\r\n" );
document.write( "Add and subtract 144 inside the parentheses:\r\n" );
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document.write( " y = -1[x² - 24x + 144 - 144]\r\n" );
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document.write( "Factor the first three terms in the brackets as the square of a\r\n" );
document.write( "binomial\r\n" );
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document.write( " y = -1[(x - 12)² - 144]\r\n" );
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document.write( "Remove the bracket leving the paretheses intact:\r\n" );
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document.write( "y = -1(x - 12)² + 144\r\n" );
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document.write( "Compare to\r\n" );
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document.write( "y = a(x-h)² + k\r\n" );
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document.write( "Vertex = (h,k) = (12,144)\r\n" );
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document.write( "Method 3 for finding the vertex:\r\n" );
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document.write( "Use the vertex formula:\r\n" );
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document.write( "The vertex is (h,k) where\r\n" );
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document.write( "h = 
and k = 
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document.write( " y = -x² + 24x\r\n" );
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document.write( " y = -x² + 24x + 0\r\n" );
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document.write( "a = -1, b = 24, c = 0\r\n" );
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document.write( "h = and k = \r\n" );
document.write( "h = 
= -12\r\n" );
document.write( "k = \r\n" );
document.write( "k = -1(12)^2+24(12)+0\r\n" );
document.write( "k = -(144)+ 288\r\n" );
document.write( "k = 144\r\n" );
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document.write( "So the vertex is (12,144)\r\n" );
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document.write( "Use whichever method for finding the vertex your teacher\r\n" );
document.write( "expects you to use, not necessarily the easiest way.\r\n" );
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document.write( "Edwin
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