document.write( "Question 1114405: A dog owner wants to enclose a rectangular area and has $840 to spend on fencing. She wants the side of the lot facing the road to have a fancier fencing material costing $18 per foot and the other three sides to use a cheaper fencing material costing $6 per foot. What is the maximum area that can be enclosed? What are the dimensions of the lot? \n" ); document.write( "

Algebra.Com's Answer #729347 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Let x be the length of the side facing the road.\r\n" );
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document.write( "and let y be the length of the adjacent side.\r\n" );
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document.write( "Then the cost of the fence is 18x + 6x + 6y + 6y = 24x + 12y.\r\n" );
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document.write( "Thus we need to maximize the area  xy  under the condition  24x + 12y = 840.\r\n" );
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document.write( "Or, which is EQUIVALENT, to maximize xy under the condition 2x + y = 70.\r\n" );
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document.write( "Express y = 70 -2x from the condition. Then we need maximize this quadratic function\r\n" );
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document.write( "f(x) = x*(70-2x) = -2x^2 + 70x.\r\n" );
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document.write( "The roots of this quadratic function are x= 0  and x= 35,  so the maximum of the quadratic function is achieved \r\n" );
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document.write( "    at the midpoint between the roots  x=  = 17.5.\r\n" );
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document.write( "Answer.  The facing to the road side must be 17.5 ft long.\r\n" );
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document.write( "         The adjacent side must be  y= 70-2x = 70 - 2*17.5 = 35 ft long.\r\n" );
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document.write( "         Then the area is 17.5*35 square ft  and the cost (Check !)  = 18*17.5 + 6*17.5 + 35*6 + 35*6 = 840 dollars.   ! Correct !\r\n" );
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