A farmer has 240 meters of fencing material to enclose his rectangular plot of land. However, it is bound on one side by a river, and on the opposite side, half the fencing is purchased and supplied by the farmer that owns the lot. See below?

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What is the maximum area of the farmer can make with this amount of fencing material and what will be the dimensions?

1 Answer
Sep 24, 2016

Let #x# be the length and #y# the width.

The perimeter is given by:

#P = x + x/2 + 2y#

#240 = x + x/2 + 2y#

#240 = (2x + x + 4y)/2#

#480 = 3x + 4y#

#-3/4x + 120 = y#

The area is given by:

A = xy

#A = (-3/4x + 120)(x)#

#A = -3/4x^2 + 120x#

The maximum area will be the vertex:

#A = -3/4(x^2 - 160x + 6400 - 6400)#

#A = -3/4(x^2 - 160x + 6400) + 4800#

#A = -3/4(x - 80)^2 + 4800#

The vertex is at #(80, 4800)#.

Hence, the maximum area is given by dimensions of #80# meters by #60# meters, where the side cut in half is #80# metres. The maximum area is #4800#.

Hopefully this helps!