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

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 $\frac{a}{2} - x$ and area would be $x \left(\frac{a}{2} - x\right)$ or $- {x}^{2} + a \frac{x}{2}$. The function will be zero when first derivative of the function is equal to zero and second derivative is negative,

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

Hence if perimeter is 800 yards and it is a square, one side would be $\frac{800}{4} = 200$ yards.

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