Answer :

ArchimedesEleven
The maximum area of a rectangle when given perimeter is simply perimeter/sides.
To get the maximum area, simply divide 160 (perimeter) by 4 (sides in a rectangle) and then square the result:
160/4 = 40
40 x 40 = 1600
The maximum area, when the sides are all of the same length, is 1600 square yards.
Hope this helps!
~ArchimedesEleven

   Maximum area covered by 160 yards fence will be 1600 square yards.

  Let the length and the width of a rectangular region are 'l' and 'w'.

Since, length of a rope covering the rectangular region is 160 yards,

Therefore, length of the rope will represent the perimeter of the rectangular region.

2(l + w) = 160

l + w = 80 ---------(1)

Expression for the area of a rectangular region is,

Area = Length × Width

A = l × w

Substitute the measure of 'w' from equation (1),

w = 80 - l

A = l(80 - l)

A = 80l - l²

For maximum area to be covered, find the derivative of the area with respect to length and equate it to zero.

A'= 80 - 2l

For A' = 0,

80 - 2l = 0

l = 40 yards

By substituting l = 40 in the expression of the area,

Area of the rectangular region = 80(40) - (40)²

                                                   = 3200 - 1600

                                                   = 1600 square yards

    Therefore, maximum region covered by 160 yards fence will be 1600 square yards.

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