Question

Use calculus to find the largest possible area for a rectangular field that can be enclosed with a fence that is 500 meters long.

Answer 1

Let $x$ and $y$ be the sides of the rectangle. The perimeter is given to be 500m, so we are maximizing the area function $A(x,y)=xy$ subject to the constraint $2x+2y=500$.

From the constraint, we find

$2x+2y=500\implies x+y=250\implies y=250-x$

so we can write the area function independently of $y$:

$A(x,y)=\hat A(x)=x(250-x)=250x-x^2$

Differentiating and setting equal to zero, we find one critical point:

$\dfrac{\mathrm d\hat A(x)}{\mathrm dx}=250-2x=0\implies x=125$

which means $y=250-125=125$, so in fact the largest area is achieved with a square fence that surrounds an area of $A(125,125)=125^2=15625\text{ m}^2$.