a circular observation window is placed in the wall of a marine biology pool as shown in the figure…

Explanation:

Step1: Identify the depth of the centroid of the window

The centroid of a circular window is at its center. The depth $h$ of the centroid of the circular window from the water - surface is $h = 10+4$ (radius of the circle is $r = 4$ ft as diameter is 8 ft), so $h=14$ ft.

Step2: Use the hydro - static pressure formula

The hydro - static pressure formula is $P=\rho gh$, where $\rho$ is the density of the fluid, $g$ is the acceleration due to gravity. In English units, when we use $\rho$ in $\text{lb/ft}^3$, the formula for pressure force on a submerged plane surface of area $A$ is $F = P\times A=\rho hA$. The area of a circle is $A=\pi r^{2}$, with $r = 4$ ft, so $A=\pi\times4^{2}=16\pi$ ft². Given $\rho = 64$ lb/ft³ and $h = 14$ ft. $F=\rho hA=64\times14\times16\pi$.

Step3: Calculate the value

$F = 64\times14\times16\pi=64\times14\times16\times3.14159$. $F=64\times14\times50.26544$. $F = 64\times703.71616$. $F\approx45038$ lb.

Answer:

45038