Answer to Question #227824 in Algebra for G12

Question

A drain has a rectangular profile. W is the width of the drain in meters, and

D is the depth of the water in meters.

In the case that the depth is much less than the width the amount of water that flows per second through a rectangular drain is given by the following formula:

Q =

A

n

D

2

3 S

1

2 (1)

In which:

- Q is the volume of water that

flows per second through the

drain (in m3/s);

- A = W×D, the cross-sectional

area of the drain up to the level

of the water (in m2)

- n is a parameter describing the

resistance to the flow of water;

- S is the gradient of the river

(in m/m).

Use formula 1 to derive the units of n.

Answer 1

$Q = \frac{A}{n} D^{2/3} S^{1/2}$

Write equality for dimentions:

$[\frac{m^3}{s}] = \frac{[m^2]}{[n]} [m]^{2/3} [\frac{m}{m}]^{1/2}$

$[\frac{m^3}{s}] = \frac{[m^{8/3}]}{[n]}$

So units of n are $[n] = [\frac{s}{m^{1/3}}]$