Chapter 4.3, Problem 19PPE

To determine

To Find :The number of cement bags needed for 4 inches radius, 5 inches radius and 6 inches radius.

Answer to Problem 19PPE

To fill a tube with 4 inches radius approximately 2 bags are needed.

To fill a tube with 5 inches radius approximately 3 bags are needed.

To fill a tube with 6 inches radius approximately 5 bags are needed.

Explanation of Solution

Given information:

Circular base area is $\pi r²$ .

Volume of single bag= $\frac{2}{3}$ feet cube

Area of tube with circular base = $\pi r²$.

Now, volume V of the tube

  $\begin{array}{l}V=l\pi r²\hfill \\ =4\pi r²\hfill \end{array}$

Volume of single bag = $\frac{2}{3}$ feet cube

Number of bags for $4\pi r²$ feet cube of volume:

  $\begin{array}{cc}y& =4\pi r^2\times \frac{3}{2}\\ & =\frac{12\pi r^2}{2}\\ & =6\pi r^2\end{array}$

Since 1 feet =12 inches

Consider $r=4$ inches. Thus,

  $\begin{array}{cc}r& =\frac{4}{12}\\ & =\frac{1}{3}\end{array}$

Thus, $r=\frac{1}{3}$ feet.

Substitute $r=\frac{1}{3}$ in this function rule and solve.

  $\begin{array}{l}6\pi r^2=6\pi {\left(\frac{1}{3}\right)}^2\\ \begin{array}{l}=6\pi \frac{1}{9}\hfill \\ =\frac{2}{3}\pi \hfill \\ =2.09\hfill \end{array}\end{array}$

Number of bags required to fill the tube with 4 inches = 2.

Consider $r=5$ inches.

Thus, $r=\frac{5}{12}$ feet.

Substitute $r=\frac{5}{12}$ in this function rule and solve.

  $\begin{array}{l}6\pi r^2=6\pi {\left(\frac{5}{12}\right)}^2\\ \begin{array}{l}=6\pi \frac{25}{144}\hfill \\ =\frac{25}{24}\pi \hfill \\ =3.27\hfill \end{array}\end{array}$

Thus, to fill a tube with 5 inches radius approximately 3 bags are needed.

Consider $r=6$ inches. Thus,

  $\begin{array}{cc}r& =\frac{6}{12}\\ & =\frac{1}{2}\end{array}$

Thus, $r=\frac{1}{2}$ feet.

Substitute $r=\frac{1}{2}$ in this function rule and solve.

  $\begin{array}{l}6\pi r^2=6\pi {\left(\frac{1}{2}\right)}^2\\ \begin{array}{l}=6\pi \frac{1}{4}\hfill \\ =\frac{3}{2}\pi \hfill \\ =4.71\hfill \end{array}\end{array}$

Thus, to fill a tube with 6 inches radius approximately 5 bags are needed.