Question

Set up and solve definite integrals to answer each of the following questions.

Find the hydrostatic force exerted on one of the long sides of a rectangular swimming pool that is $20$feet long, 12 feet wide, and $6$feet deep.

Short answer

$269,568pounds.$

Step-by-step solution

Step 1. Given Information.

The hydrostatic force exerted on one of the long sides of a rectangular swimming pool that is $20$feet long, $12$feet wide, and $6$feet deep.

Step 2. Diagram.

Consider that the top of the tank is at height $y=6$and the bottomof the tank is at height $y=0$. Draw a diagram that shows a thin representative slice of the tank at some point $y_k^{*}$from the bottom.

Step 2. Formulation.

Assume that the entire thin slice of wall is at a depth of $d_k=6-y_k^{*}units$.

The area of the representative wall slic is $A_k=240∆y$square feet.

The water density is $\omega =62.4$pounds per cubic foot.

The hydrostatic force exerted by a water of weight-density $\omega$and depth density $d$on a horizontal line of area $A$is given by

$F=\omega Ad$.

Substitute , $A_k=240∆y,d_k=6-y_k^{*},\omega =62.4$in $F=\omega Ad$to obtain

$F=62.4\left(6-y_k^{*}\right)240∆y$.

Step 4. Calculation.

The hydostatic force on the entire side wall is approximately

$F=\sum _{k=1}^{n}62.4\left(6-y_k^{*}\right)240∆y$

As $n\to \infty$, $F=\sum _{k=1}^{n}62.4\left(6-y_k^{*}\right)240∆y$becomes a definite integral.

Accumulate the slices from $y=0toy=6$in order to obtain the hydostatic force on the entire side wall.

$$\begin{gathered} W={\int }_0^{6}62.4\left(240\right)\left(6-y\right)dy \\ W=62.4\left(240\right){\int }_0^6\left(6-y\right)dy \\ W=62.4\left(240\right){\left(6y-\frac{y^2}{2}\right)}_0^6 \\ W=269,568 \end{gathered}$$.