document.write( "Question 1192493: Three sides of a trapezoid are each 10 cm long. How long is the 4th side, when the area of the trapezoid has the greatest value? \n" ); document.write( "

Algebra.Com's Answer #824374 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Three sides of a trapezoid are each 10 cm long.
\n" ); document.write( "How long is the 4th side, when the area of the trapezoid has the greatest value?
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document.write( "It is clear, that under given conditions, the trapezoid is isosceles with the shorter base of 10 cm long.\r\n" );
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document.write( "See my sketch below.  The upper base is 10 cm;  the lower base is 10+2x cm long.\r\n" );
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document.write( "                  10\r\n" );
document.write( "         +------------------+\r\n" );
document.write( "        /|                   \\r\n" );
document.write( "       / |                    \\r\n" );
document.write( "  10  /  |                     \  10\r\n" );
document.write( "     /   |                      \\r\n" );
document.write( "    /    |                       \\r\n" );
document.write( "   +------------------------------+\r\n" );
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document.write( "   <- x ->\r\n" );
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document.write( "The mid-line is   = 10+x  cm long.\r\n" );
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document.write( "The height is  .\r\n" );
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document.write( "The area of the trapezoid  is   A(x) = .\r\n" );
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document.write( "To find the maximum of A(x), take the derivative\r\n" );
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document.write( "     =  - \r\n" );
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document.write( "     =  - \r\n" );
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document.write( "Equate it to zero\r\n" );
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document.write( "     -  = 0.\r\n" );
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document.write( "Simplify and find x\r\n" );
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document.write( "     = \r\n" );
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document.write( "     = (10+x)*x\r\n" );
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document.write( "     = \r\n" );
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document.write( "     = 0\r\n" );
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document.write( "     = 0\r\n" );
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document.write( "    (x-5)*(x+10) = 0\r\n" );
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document.write( "Of two roots,  x= 5 and x= -10, take the positive one, which gives the maximum to the trapezoid's area.\r\n" );
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document.write( "ANSWER.  The area of trapezoid is maximum when the 4-th side is 10+2*5 = 20 cm long.\r\n" );
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