document.write( "Question 1146864: A window is to be constructed in the shape of an equilateral triangle on top of a rectangle if its perimeter is to be 600 cm, What is the maximum possible area of the window?
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Algebra.Com's Answer #768162 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Let W be the width of the window, in centimeters.\r\n" );
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document.write( "Then the vertical length of the rectangular part of the window is L = 0.5*(600 - 3W) centimeters.\r\n" );
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document.write( "Thus the area of this special form window is\r\n" );
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document.write( "     A(w) = W*0.5*(600-3W) +   cm^2.     (1)\r\n" );
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document.write( "The first addend in the formula is the area of the rectangular part, while the second addend is the area of the triangular part.\r\n" );
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document.write( "Thus the area is this quadratic function of the variable \"w\"\r\n" );
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document.write( "    A(w) =  +  +  =  + 300*W.    (2)\r\n" );
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document.write( "They ask to find the maximum of this quadratic form.\r\n" );
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document.write( "For the general quadratic form f(x) = ax^2 + bx + c  with negative leading coefficient \"a\",  \r\n" );
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document.write( "the maximum is achieved at x = .\r\n" );
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document.write( "In our case,  a = ,  b = 300.\r\n" );
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document.write( "Therefore, the quadratic form achieves the maximum at\r\n" );
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document.write( "    W =  = 140.583 cm.\r\n" );
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document.write( "To find the maximum area, simply substitute this value of W into the formula (2)\r\n" );
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document.write( "     =  + 300*140.583 = 21087.41 cm^2.      ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "On finding maximum/minimum of a quadratic functions and solving other similar minimax problems see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - A rectangle with a given perimeter which has the maximal area is a square\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular garden to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A farmer planning to fence a rectangular area along the river to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area\r
\n" ); document.write( "\n" ); document.write( "    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function\r
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\n" ); document.write( "\n" ); document.write( "A convenient place to observe all these lessons from the \"bird flight height\" is the last lesson in the list, marked by (*).\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this textbook under the topic \"Finding minimum/maximum of quadratic functions\". \r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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