How can I write and solve an equation to find the depth at which the pressure is 200 kN/m?

Question

A hydraulic engineer student at University of Hawaii West Oahu is studying the pressure in a particular fluid. The pressure is equal to the atmospheric pressure 101 kN/m plus 8 kN/m for every meter below the surface, where kN/m is kilonewtons per meter, a unit of pressure.

Answer 1

Let P(x) = the fluid pressure at depth x, where x = the number of meters below the surface. If x=0 equates to the surface of the fluid, and we measure depth with negative values of x (e.g. x = -2 is 2 meters below the surface), then

 

P(x) = (101 kN/n) - (8 kN/m)x

 

This is a linear equation with a slope of -8 and a y-intercept of 101.  The domain of this function is x ≤ 0.

Answer 2

The equation is

 

P = 101 + 8 d; Assume the depth measured below is positive (to keep it simple)

 

200 = 101 + 8 d

 

subtract 101 from both side

 

99 = 8 d

 

Divide by 8 on both side

 

d = 99/8 = 12 3/8 feet = 12 feet 4.5 inches

Answer 3

The pressure is atmospheric pressure (101 kN/m^2) + 8 kN/m^3 (d m)

 

p = 101 + 8 d

 

where d is the depth in m from the surface.  If you consider a coordinate system which is positive upward, then the equation can be written as

 

p = 101 - 8 d; for d < 0 indicating below the origin.

 

If written in absolute values, the same can be written as

 

p = 101 + 8 |d|