Chapter 6, Problem 56RE

To determine

Cubic yards of dirt required to fill the area after the swamp is drained.

Answer to Problem 56RE

1500 cubic yards of dirt is required to fill the area after the swamp is drained.

Explanation of Solution

Given information:

  

Average depth,

  ${\text{D}}_{avg}=5\text{ft}$

Since the pond has been divided into six sections, which is even, approximate the surface area of the pond using Simpson’s rule.

We know that

Horizontal spacing is 20 ft.

Then let

  $h=20$

Using Simpson’s rule:

  $\begin{array}{l}S=\frac{h}{3}\left(y_0+4y_1+2y_2+4y_3+2y_4+4y_5+y_6\right)\\ =\frac{20}{3}\left(146+4\left(122\right)+2\left(76\right)+4\left(54\right)+2\left(40\right)+4\left(30\right)+13\right)\\ =\frac{20}{3}\left(1215\right)\\ =8100{\text{ft}}^{\text{2}}\end{array}$

Now,

To find the volume in cubic feet, multiply the volume obtained with average depth:

  $\begin{array}{l}V=S\times {\text{D}}_{Avg}\\ =8100{\text{ft}}^{\text{2}}\times 5\text{ft}\\ =\text{40,500}{\text{ft}}^3\end{array}$

We know that

  $1\text{yd}=3\text{ft}$

For 1 cubic yard:

  $1{\text{yd}}^3={\left(3\text{ft}\right)}^3=27{\text{ft}}^{\text{3}}$

Now,

To get the volume in cubic yards, divide the volume in cubic feet by $27{\text{ft}}^{\text{3}}$ :

  $V=\frac{40,500}{27}=1500{\text{yd}}^{\text{3}}$

Thus,

1500 cubic yards of dirt is required to fill the area after the swamp is drained.