Question

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

Answer 1

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.