document.write( "Question 1139604: Peter’s farm has 160 meters of fencing, and he wants to fence a rectangular field that borders a straight river. He needs no fence along the river side. Find the largest area of Peter’s farm that can be fenced. \n" ); document.write( "

Algebra.Com's Answer #760084 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "Since one side is the river, the rectangle's fence perimeter will be\r\n" );
document.write( "\r\n" );
document.write( "L + 2W = 160.\r\n" );
document.write( "\r\n" );
document.write( "Hence, L = 160 - 2W.\r\n" );
document.write( "\r\n" );
document.write( "Area = Length * Width.\r\n" );
document.write( "\r\n" );
document.write( "Substitute (160-2W) for L:\r\n" );
document.write( "\r\n" );
document.write( "A = W(160 - 2W)\r\n" );
document.write( "\r\n" );
document.write( "A = -2W^2 + 160W.\r\n" );
document.write( "\r\n" );
document.write( "This is a quadratic function. It has the maximum at x = -b/(2a), according to the general theory.\r\n" );
document.write( "\r\n" );
document.write( "    (See the lessons\r\n" );
document.write( "     \r\n" );
document.write( "         - HOW TO complete the square to find the minimum/maximum of a quadratic function\r\n" );
document.write( "\r\n" );
document.write( "         - Briefly on finding the minimum/maximum of a quadratic function\r\n" );
document.write( "\r\n" );
document.write( "     in this site).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For our quadratic function the maximum is at\r\n" );
document.write( "\r\n" );
document.write( "W =  =  = 40.\r\n" );
document.write( "\r\n" );
document.write( "So, W = 40 meters is the width for max area.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the length is  L = 160 - 2W = 160 - 2*40 = 80 meters.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the maximal area is L*W = 80*40 = 3200 square meters.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The plot of the quadratic function  y = - 2x^2 + 160x  for the area is shown below:  y = area and x = width.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Plot y = -2x^2 + 160x.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );