Rectangular pen along a river

Text

A rectangular plot is to be fenced with $120\text{ m}$ of fencing; one side lies along a river and needs no fence. What dimensions maximize the enclosed area?

Solution

Let $x$ be the two sides perpendicular to the river and $y$ the parallel side. Constraint: $2x+y=120$, so $y=120-2x$. Area: $A(x)=120x-2x^{2}$; $A'(x)=120-4x=0\Rightarrow x=30$, hence $y=60$. $A''<0$: maximum. $A=1800\text{ m}^2$. $\boxed{x=30\ \text{m},\ y=60\ \text{m},\ A_{\max}=1800\ \text{m}^2}$