You have a straight line of length $b$. You want to connect the ends of this fence so as to enclose a maximum area. You have a cost constraint. In the area between $x=0$ and $x=\frac{b}{2}$ costs $1$ dollar per ft and from $x=\frac{b}{2}$ to $x=b$ costs $2$ dollars per ft.
I realize I need to use lagrange multipliers and arc length. could someone get me started on this problem. any help or links would be appreciated..
Assume your fence runs from $O=(0,0)$ to $P=\bigl({b\over2},h\bigr)$ and then to $Q=(b,0)$ and that the arc length between $O$ and $P$ is $s_1$, between $P$ and $Q$ is $s_2$. Then the solution to Dido's problem tells you that the enclosed area between the base $OQ$ and the fence is largest when the two arcs are circular arcs of the given lengths connecting $O$ and $P$, resp., $P$ and $Q$. In this way your variational problem is reduced to a problem in a finite number of variables.