Problem $8$

A steady stream of air is blowing downward on a triangular hut as shown. The hut is $10m$ long into the page and $3m$ high, measured perpendicularly from the ground to the apex. The two side walls of the hut meet at a $90$ angle. The pressure distribution on the hut is symmetric about the $y$ axis and is given by$,$

$P(r)=P_{\text{a}\text{t}\text{m}}+\frac{}{2}(U_0^2-A{r}^{\frac{2}{3}})$

where $r$ is measured from the apex along a side of the hut, $U_0=3\frac{m}{s},A=10\frac{m^{\frac{4}{3}}}{s^2},$ and $=1\mathrm{.}2k\frac{g}{m^3}.$ The air inside the hut is uniformly at atmospheric pressure. Calculate the minimum mass of the hut required to keep it on the ground.