You have 200 yards of fencing and wish to enclose a rectangular area.

Create a function such that it expresses width of the rectangular area.

The perimeter of any given rectangular can be found with: $${2l+2w=perimeter}$$

$${2l+2w=200}$$ $${2l=-2w+200}$$ $${l=\frac{-2w+200}{2}}$$ $${l=-w+100}$$

Now that we know the length find the area with $${a=lw}$$

$${a(w)=(-w+100)w}$$ $${a(w)=-w^2+100w}$$

When is the area the largest for $$w$$

To find the maxima of any given graph use $${\frac{-b}{2(a)}}$$

$$a=-1$$ $$b=100$$

$${\frac{-(100)}{2(-1)}}$$ $${\frac{-100}{-2}}$$ $${maxima=50}$$

Find the maximum area of the rectangle

Now that you know that maximum width you can find the maximum area for the rectangle by using the maximum as an input to the function $$a(w)$$

$$a(50)$$ $${a(w)=-w^2+100w}$$ $${a(50)=-50^2+100(50)}$$ $${a(50)=-2500+5000}$$ $${a(50)=2500}$$