Liana has 800 yards of fencing to enclose a rectangular area. How do you maximize the area?

1 Answer
Shwetank Mauria
Feb 19, 2016

Area can be maximized by fencing a square of side #200# yards.

Explanation:

Given perimeter of a rectangle, square has the maximum area (proof given below).

Let #x# be one of the side and #a# be te perimeter then the other side would be #a/2-x# and area would be #x(a/2-x)# or #-x^2+ax/2#. The function will be zero when first derivative of the function is equal to zero and second derivative is negative,

As first derivative is #-2x+a/2# and this will be zero, when #-2x+a/2=0# or #x=a/4#. Note that second derivative is #-2#. Then two sides will be #a/4# each that the it would be square.

Hence if perimeter is 800 yards and it is a square, one side would be #800/4=200# yards.

Hence area can be maximized by fencing a square of side #200# yards.

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