= Minimum roof length at fixed wall height and cross-sectional area
{title2=$2\tan2\theta=w/h,\quad w^4=16A(A-wh)$}
For $A,h>0$, a rectangular cross section of width $w$ and height $h$, capped by a symmetric triangular roof at angle $0<\theta<\pi/2$, has area $A=wh+w^2\tan\theta/4$. Its total sloping roof length is $w\sec\theta$. A cost $2\alpha h+\beta w\sec\theta$ with $\beta>0$ has the same minimizing shape as this roof length, since the fixed wall term is constant.
The <Lagrange multiplier> equations for $w\sec\theta$ under the area constraint give $w(1-\tan^2\theta)=4h\tan\theta$. This is the first displayed condition; it also implies $\theta<\pi/4$. Eliminating $\tan\theta=4(A-wh)/w^2$ gives the second condition.
For a global check, the squared roof length is $Q(w)=w^2+16(A-wh)^2/w^2$ for $0<w<A/h$. Its <derivative> has the sign of $w^4+16Ahw-16A^2$, a strictly increasing expression with one root in that interval. Thus this root is the unique minimum. The shape is independent of the positive roof-cost coefficient and the constant wall cost. If $\beta=0$, every feasible shape has the same cost and these conditions are no longer necessary.
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