Q. 65 - Calculus

Question

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

The hydrostatic force exerted by the water in a birdbath on its cylindrical side, given that the birdbath is in the shape of a cylinder that is 1 foot deep with a radius of 0.4 foot.

Step-by-Step Solution

Answer

The hydrostatic force exerted by the water in a birdbath on its cylindrical side is $2$ square feet.

Step 1. Given Information.

The birdbath is in the shape of a cylinder that is 1 foot deep with a radius of 0.4 foot.

Step 2. Find A k , d k

Area of the cylinder$$\begin{gathered} A_k=2\mathrm{\pi }(0.4)\left(1\right) \\ =0.8\mathrm{\pi } \end{gathered}$$

Depth, $d_k=0.4-x_k^{*}$

The density of the water is $62.4$ pounds.

Step 3. Find the hydrostatic force.

The hydrostatic force is given by, 

$$\begin{gathered} F=\omega A_K{d}_k \\ =62.4(0.8\mathrm{\pi }∆\mathrm{y})0.4-{\mathrm{y}}_{\mathrm{k}}^{*} \\ =\sum _{\mathrm{k}=1}^{\mathrm{n}}62.4(0.8\mathrm{\pi })0.4-{\mathrm{y}}_{\mathrm{k}}^{*}(∆\mathrm{y}) \\ \mathrm{F}={\int }_0^{0.4}62.4(0.8(3.14\left)\right)[0.4-\mathrm{y}]\mathrm{dy} \\ =24.96\mathrm{\pi }{[0.4\mathrm{y}-\frac{{\mathrm{y}}^2}{2}]}_0^{0.4} =24.96\mathrm{\pi }[0.08] \\ \simeq 2 \end{gathered}$$