Hydrostatic pressure problem...

[MarcC]

Determine the force due to hydrostatic pressure on the flat vertical side of a tank which has the shape in feet of the boundaries y=0, y = 8 ln x, y = 8 ln(−x) and the line y=8 ln18.
Note that water has density 62.4 lb/ft3.

Can somebody help me with this? My problem is that I have absolutely no idea where to start (and the textbook is not helping matters...)

 

[DrPhil]

Determine the force due to hydrostatic pressure on the flat vertical side of a tank which has the shape in feet of the boundaries y=0, y = 8 ln x, y = 8 ln(−x) and the line y=8 ln18.
Note that water has density 62.4 lb/ft3.

Can somebody help me with this? My problem is that I have absolutely no idea where to start (and the textbook is not helping matters...)

The tank is a volume bounded by surfaces. One of the bounding surfaces is a vertical plane, and the boundaries of that surface are four curves or lines in the xy-plane.
IT IS NOT CLEAR WHICH IS THE TOP AND WHICH IS THE BOTTOM, but in the absence of that bit of information I will assume that
$\displaystyle y_1=0$ is the bottom, and
$\displaystyle y_2=8 \ln 18$ is the top.
Assuming also that the tank is full, we will find the total force by integrating the incremental Force
$\displaystyle \displaystyle \int_{y_1}^{y_2} F(y)\ dy$.

How wide is the tank at a height $\displaystyle y$ above the bottom? It looks symmetric in the x-direction, with the two side given by
$\displaystyle \displaystyle x = -e ^{y/8} \text{ and }x = +e ^{y/8}$

What is the incremental Area (square feet) between $\displaystyle y$ and $\displaystyle y+dy$?

Find the pressure $\displaystyle P(y)$ as the weight of a 1-sq.ft. column of water above $\displaystyle y$, and
use the relationship $\displaystyle F = P \times A$ to find the incremental force.

If you need more help, please show us your work.