An equation that is commonly used to describe the volume flow rate of water in an open channel is given by
$$
Q=\frac{1}{n} \frac{A^{\frac{5}{3}}}{P^{\frac{2}{3}}} S_{0}^{\frac{1}{2}}
$$
where $Q$ is the volume flow rate in the channel $\left[\mathrm{L}^{3} \mathrm{~T}^{-1}\right], n$ is a constant that characterizes the roughness of the channel surface [dimensionless], $A$ is the flow area $\left[\mathrm{L}^{2}\right]$, $P$ is the perimeter of the flow area that is in contact with the channel boundary [L], and $S_{0}$ is the slope of the channel [dimensionless]. This equation is usually applied using SI units, where $Q$ is in $\mathrm{m}^{3} / \mathrm{s}, A$ is in $\mathrm{m}^{2},$ and $P$ is in $\mathrm{m}$. (a) Is the given equation dimensionally homogeneous? (b) If the equation is not dimensionally homogeneous, what conversion factor must be inserted after the equal sign for the equation to work with $Q$ in $\mathrm{ft}^{3} / \mathrm{s}, A$ in $\mathrm{ft}^{2},$ and $P$ in $\mathrm{ft}$ ?
