document.write( "Question 1169039: Picture frame has its length 8cm longer than its width. It has an inner 1-cm boundary such that a maximum 660cm^2- picture may fit into it. Find the dimension of this frame. \n" ); document.write( "

Algebra.Com's Answer #851295 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Picture frame has its length 8cm longer than its width. It has an inner 1-cm boundary
\n" ); document.write( "such that a maximum 660cm^2- picture may fit into it. Find the dimension of this frame.
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\n" ); document.write( "\n" ); document.write( " The problem formulation as it is given in the post is INCORRECT.\r
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\n" ); document.write( "\n" ); document.write( " The correct formulation, in my version, is THIS\r
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document.write( "    Picture frame has its length 8cm longer than its width. It has an inner 1-cm boundary \r\n" );
document.write( "    such that the area of the picture itself is 660cm^2. Find the dimension of this frame.\r\n" );
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\n" ); document.write( "\n" ); document.write( " Below is my solution for this edited formulation.\r
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document.write( "Since the width of the frame is uniform and since the outer length of the frame is 8 cm longer \r\n" );
document.write( "than its outer width, the length of the picture itself, without considering the frame, is\r\n" );
document.write( "8 cm longer than the picture width.\r\n" );
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document.write( "So, if x is the picture width, in centimeters, then the picture length is (x+8) cm,\r\n" );
document.write( "and we have this equation\r\n" );
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document.write( "    x*(x+8) = 660 cm^2.     (1)\r\n" );
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document.write( "Now you can guess mentally the solution to this equation: it is x= 22 cm for the picture width, \r\n" );
document.write( "giving  22+8 = 30 cm  for the picture length.\r\n" );
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document.write( "After this guessing, note that the left side of the equation (1) is the monotonic function,\r\n" );
document.write( "so the guessed solution is UNIQUE.\r\n" );
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document.write( "Alternatively, you can solve this quadratic equation (1) formally, using factoring or the quadratic formula.\r\n" );
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document.write( "In either case, you will get the same solution for x.\r\n" );
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document.write( "Thus the picture is 22 x 30 centimeters.\r\n" );
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document.write( "It implies that the outer dimensions of the frame are  24 x 32 cm.    \r\n" );
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document.write( "ANSWER.  The outer dimensions of the frame are  24 x 32 cm.    \r\n" );
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