Chapter P, Problem 26E

Manning’s Equation Hydrologists sometimes use Manning’s equation to calculate the velocity $v$ in feet per second, of water flowing through a pipe. The velocity depends on the hydraulic radius $R$ in feet, which is one-quarter of the diameter of the pipe when the pipe if flowing full; the slope $S$ of the pipe, which gives vertical drop in feet for each horizontal foot; and the roughness coefficient $n$, which depends on the material of which the pipe is made. The relationship is given by

$v=\frac{1.486}{n}{R}^{2/3}{S}^{1/2}$

For a certain brass pipe, the roughness coefficient has been measured to be $n=0.012$. The pipe has a diameter of $3$ feet and a slope of $0.2$ foot per foot. (That is, the pipe drops $0.2$ foot for each horizontal foot.) If the pipe is flowing full, find the hydraulic radius of the pipe, and find the velocity of the water flowing through the pipe.