A drainage tile is a cylindrical shell $42\text{cm}$ long. The inside and outside diameters are $8\text{cm}$ and $14\text{cm}$, respectively. What is the volume (in cm³) of clay required for the tile? (Use $\pi =\frac{22}{7}$)

The correct answer is 4356.

To find the volume of clay required to make the drainage tile, we need to calculate the volume of the cylindrical shell. A cylindrical shell is formed by an outer cylinder with a hollow inner cylinder.

The volume of a cylindrical shell, $V$, is given by the difference between the volume of the outer cylinder and the volume of the inner cylinder:
$V=\pi R^{2}h-\pi r^{2}h=\pi h(R^2-r^2)$
where:
• $h$ is the length (height) of the cylinder,
• $R$ is the outer radius, and
• $r$ is the inner radius.

Let us identify the given values from the problem statement:
• Length of the tile, $h=42\text{cm}$
• Inner diameter = $8\text{cm}$, so the inner radius $r=\frac{8}{2}=4\text{cm}$
• Outer diameter = $14\text{cm}$, so the outer radius $R=\frac{14}{2}=7\text{cm}$
• Value of $\pi$ to be used is $\frac{22}{7}$

Substitute these values into the volume formula:
$V=\frac{22}{7}\times 42\times (7^2-4^2)$

First, simplify the term outside the parenthesis:
$\frac{22}{7}\times 42=22\times 6=132$

Next, calculate the difference between the squares of the radii:
$7^2-4^2=49-16=33$

Now, multiply these two results together to find the final volume:
$V=132\times 33$

Let us perform the multiplication:
$132\times 33=132\times (30+3)=3960+396=4356$

Thus, the volume of clay required for the drainage tile is 4356 cm³.