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## Q. 1.1

A sensitive pressure transducer is used to record the pressure at the bottom of a river that carries a sediment load that causes the density to increase linearly from 1.945 slug/$ft_{3}$ on the surface at a rate of 0.012 slug/ft³/ft of depth. Develop the equation that gives the depth of flow from this pressure reading. What is the error if the sediment load is ignored, and the density taken equal to 1.94 and a pressure of 3.5 psi is recorded?

## Verified Solution

The problem is solved by defining ρ = 1.945 + 0.012h, substituting this into Equation 1.4b, separating variables, and integrating. The result is

$\frac{dp}{dh}=\gamma=\rho g$              (1.4b)

p = g(1.945h + 0.006h²),

where
p is in pound per square foot
h is in feet

Applying the quadratic formula gives the depth h as a function of the pressure reading as

$h=-162.083+83.333\left\lgroup3.783+0.024\frac{p}{g}\right\rgroup^{1/2}.$

Substituting p = 3.5 × 144 into the above equation gives h = 7.856 ft. If ρ = 1.94 (constant), then p = 62.4h, or h = 8.077 ft, or an error of +0.221 ft.