$\text {It is desired to transport $4 \mathrm {~m} ^ { 3 } / \mathrm { s }$ of water through a rectangular finished concrete canal that can}$$\text { be at most $2 \mathrm {~m}$ wide. The slope of the land through which it passes is $0.0004$. What depth of}$ $\text {water should be expected?}$
A) 1.01 m
B) 1.28 m
C) 1.44 m
D) 1.83 m

Question

Quizplus · Accepted Answer

$1.44 \mathrm {~m}$
$\text {The flow rate is given by Eq. 7.7.6:}$
$$Q = \frac { c _ { 1 } } { n } A R ^ { 1 / 3 } S ^ { 1 / 2 } \text { or } 4 = \frac { 1 } { 0.012 } \times 2 h \times \left( \frac { 2 h } { 2 h + 2 } \right) ^ { 1 / 3 } 0.0004 ^ { 1 / 2 } \text { or } h \left( \frac { h } { h + 1 } \right) ^ { 1 / 3 } = 1.2$$
$\text {This again requires a trial-and-error solution (a usual method when the flow rate is}$ $\text {unknown):}$
$$\begin{array} { l l } 
h = 1 : & 0.794 \neq 1.2 \
h = 2 : & 1.754 \neq 1.2 \
h = 1.5 : & 1.27 \neq 1.2 \
h = 1.44 : & 1.21 \neq 1.2
\end{array}$$
$\text {Select $h = \underline { 1.44 \mathrm {~m} }$.}$