Q. 63 - Calculus

Question

Find the hydrostatic force described in each of Exercises
61–69.

The hydrostatic force exerted on one of the long sides of a giant rectangular hot tub that is 5 feet wide, 8 feet long, and 3 feet deep.

Step-by-Step Solution

Answer

The hydrostatic force exerted on one of the long sides of a giant rectangular hot tub that is 5 feet wide, 8 feet long, and 3 feet deep is $4,000$ pounds.

Step 1. Given Information.

A rectangular hot tub that is 5 feet wide, 8 feet long, and 3 feet deep.

Step 2. Find A k , d k .

Area of the rectangular tub$=8∆y$

Depth, $d_k=3-y_k^{*}$

The density of the water is $\omega =62.4$ pounds.

Step 3. Find the hydrostatic force.

The hydrostatic force is given by,

$$\begin{gathered} F=\omega A_k{d}_k \\ \simeq 62.4(8∆y)3-y_k^{*} \\ =\sum _{k=1}^{n}62.4\left(8\right)(3-y_k^{*})∆y \end{gathered}$$

$$\begin{gathered} F={\int }_0^8(62.4(8\left)\right(3-y\left)\right)dy \\ =500{[3y-\frac{y^2}{2}]}_0^8 =500[-8] \\ =4000 pounds \end{gathered}$$