Chapter 5, Problem 58RE

To determine

To count number of cubic yards of dirt will it take to fill the area after the swamp is drained.

Answer to Problem 58RE

Number of cubic yards of dirt will it take to fill the area after the swamp is drained is $250$ cubic yard.

Explanation of Solution

Given:

  

Concept used:

Simpson's rule is as:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\frac{\Delta x}{3}\left[f\left(x_0\right)+4f\left(x_1\right)+2f\left(x_2\right)+\mathrm{......}+2f\left(x_{n-2}\right)+4f\left(x_{n-1}\right)+f\left(x_n\right)\right]$

Where

  $\Delta x=\frac{b-a}{n}$

Calculation:

The area of lake can be approximated using simpson's rule.

From

  

Vertical measurements

  $\begin{array}{c}n+1=7\\ n=7-1\\ =6\end{array}$

And horizontal spacing

  $b-a=20$

Hence

  $\begin{array}{c}\Delta x=\frac{20}{6}\\ =\frac{10}{3}\end{array}$

Thus, the approximated area of lake

  $\begin{array}{l}\text{Area}=\frac{10}{\frac{3}{3}}\left[146+\left(4\times 122\right)+\left(2\times 76\right)+\left(4\times 54\right)+\left(2\times 40\right)+\left(4\times 30\right)\right]\\ =\frac{10}{9}\times \left(1215\right)\\ =1350\end{array}$

Now

  $\begin{array}{l}\text{Volume}=5\times \left(1350\right)\\ =6750f{t}^3\end{array}$

Hence number of cubic yards of dirt will it take to fill the area after the swamp is drained:

  $\frac{6750}{27}=250\text{cubic}\text{yard}$

Conclusion:

Number of cubic yards of dirt will it take to fill the area after the swamp is drained is $250$ cubic yard.