Q. 24 - Calculus

Question

The hydrostatic force exerted by a liquid of weight-density 70 pounds per cubic foot on the bottom of a cylindrical tank with height 8 feet and radius 2 feet.

Step-by-Step Solution

Answer

The hydrostatic force exerted by a liquid of weight-density $70$ pounds per cubic foot on the bottom of a cylindrical tank with height $8$ feet and radius $2$ feet is 36 pounds.

Step 1. Given Information.

The given height and radius are 8 feet and 2 feet respectively The hydrostatic force exerted by a liquid of weight-density is 70 pounds per cubic foot.

Step 2. Calculate

$$\begin{gathered} F=\int P\left(y\right)L\left(y\right)dy, P\left(y\right)=y·w, L\left(y\right)=2\sqrt{1.5^2-y^2} \\ ={\int }_0^{1.5}y·w·2\sqrt{1.5^2-y^2}dy \\ =2w{\int }_0^{1.5}y\sqrt{1.5^2-y^2}dy \\ =2w{{[-\frac{1}{3}{(1.5^2-y^2)}^{\frac{3}{2}}]}_0}^{1.5} \\ =2.42(0-(-\frac{1}{3})·1.5^3) \\ =2.42·\frac{1}{3}·1.5^3 \\ =36pd \end{gathered}$$