Q. 67 - Calculus

Question

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

The hydrostatic force exerted on a dam in the shape of a trapezoid whose top is 400 feet long and whose base is 250 feet long, given that the dam is holding back a body of water that reaches all 200 feet from the bottom to the top of the dam.

Step-by-Step Solution

Answer

The hydrostatic force exerted on a dam in the shape of a trapezoid is $405,600,000$.

Step 1. Given Information.

A trapezoid top is 400 feet long and whose base is 250 feet long, given that the dam is holding back a body of water that reaches all 200 feet

Step 2. Find A k , d k

Area of the trapezoid$$\begin{gathered} A_k=\frac{(400+250)}{2}∆y \\ =325∆y \end{gathered}$$

Depth, $d_k=200-y_k^{*}$

The density of the water is $62.4$ pounds.

Step 3. Find the hydrostatic force.

The hydrostatic force is given by, 

$$\begin{gathered} F=\omega A_k{d}_k \\ =62.4\left(325\right)(200-y_k^{*})∆y \\ F=20280{\int }_0^{200}(200-y)dy \\ =20,280{[200y-\frac{y^2}{2}]}_0^{200} =20,280[20,000] \\ =405,600,000 \end{gathered}$$