Q. 66 - Calculus

Question

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

The hydrostatic force exerted on a dam in the shape of an isosceles triangle whose top is 200 feet wide and whose total height is 100 feet, given that the dam is holding back a body of water that reaches to the top of the dam.

Step-by-Step Solution

Answer

The hydrostatic force exerted on a dam in the shape of an isosceles triangle is $31,200,000$.

Step 1. Given Information.

An isoseles triangle whose top is 200 feet wide and whose total height is 100 feet

Step 2. Find A k , d k

Area of the isosceles triangle$$\begin{gathered} A_k=\frac{1}{2}\times 200∆y \\ =100∆y \end{gathered}$$

Depth, $d_k=100-y_k^{*}$

The density of the water is $62.4$ pounds.

Step 3. Find the hydrostatic force.

The hydrostatic force is given by, 

$$\begin{gathered} F=\omega A_k{d}_k \\ =62.4\left(100\right)(100-y_k^{*})∆y \\ F=6240{\int }_0^{100}(100-y_k^{*})∆y \\ =6240{[100y-\frac{y^2}{2}]}_0^{100} =6240\left[5000\right] \\ =31,200,000 \end{gathered}$$