If a parabolic cable $($zero bending stiffness$)$ under uniform load is inverted, the shape is a funicular arch $($see figure$).$ What is the optimal height, $h,$ that minimizes the weight of the cable, when the design is done under conditions of a maximum $($constant$)$ allowable stress ${}_0?$

Note: The area of the cable is given by $A=\frac{F}{{}_0},$ where $F$ is the resultant force $($axial$)$ at any point $x.$