jmc

Let the length of the backyard be $l$ and the width be $w$. We have the equation $l+2w=360$. We want to maximize the area of this rectangular backyard, which is given by $lw$. From our equation, we know that $l=360-2w$. Substituting this into our expression for area, we have $$(360-2w)(w)=360w-2w^2$$We will now complete the square to find the maximum value of this expression. Factoring a $-2$ out, we have $$-2(w^2-180w)$$In order for the expression inside the parenthesis to be a perfect square, we need to add and subtract $(180/2{)}^2=8100$ inside the parenthesis. Doing this, we get $$-2(w^2-180w+8100-8100)\Rightarrow -2(w-90{)}^2+16200$$Since the maximum value of $-2(w-90{)}^2$ is 0 (perfect squares are always nonnegative), the maximum value of the entire expression is 16200, which is achieved when $w=90$ and $l=360-2w=180$. Thus, the maximum area of the backyard is $\boxed{16200}$ square feet.