Chapter 7.5, Problem 4QQ

a.

To determine

To find: The Reiman Sum approximating the force exerted on the entire front of the tank.

Answer to Problem 4QQ

The required format sum is, ${\displaystyle \sum _{i=1}^{n}62.4h_1\times 2\Delta h}$ .

Explanation of Solution

Given information:

The figure is:

  

The pressure given as $\rho =62.h;W=2\text{ft; }h=1.5\text{ft}$ where, W is the width and h is the height

Formula used:

  $\text{Force}=\text{Pressure}\times \text{Area}$ .

Calculation:

  $\text{Force}={\displaystyle \sum _{i=1}^{n}62.4h_1\times 2\Delta h}$ .

b.

To determine

To find a definite integral using the obtained figure in part a.

Answer to Problem 4QQ

The force is ${\displaystyle {\int }_0^{1.5}\left(62.4h\times 2dh\right)}$ .

Explanation of Solution

Given information: The Reiman sum obtained from the previous part is $\text{Force}={\displaystyle \sum _{i=1}^{n}62.4h_1\times 2\Delta h}$ .

Calculation:

From the Reiman Sum it can be written as, ${\displaystyle {\int }_0^{1.5}\left(62.4h\times 2dh\right)}$

c.

To determine

To find the total force exerted on the front of the tank if the front and back are semi circles with a given diameter.

Answer to Problem 4QQ

The total force is 41.6 lbs.

Explanation of Solution

Given information:

The Reiman sum is previously obtained as ${\displaystyle {\int }_0^{1.5}\left(62.4h\times 2dh\right)}$ .

Calculation:

The equation for force ill change as the area is changed.

Given diameter is 2 ft. So, radius will be 1 ft.

The new equation will be $F=62.4h_{i}2\sqrt{1^2-h^2}\Delta h$ .

The total force exerted will be:

  $\begin{array}{c}{\displaystyle {\int }_0^{1}62.4h\left(2\sqrt{1^2-h^2}\right)dh}=2\left(62.4\right){\displaystyle {\int }_0^{1}h\left(\sqrt{1^2-h^2}\right)dh}\\ =41.6\end{array}$