Question

A well of inner diameter 14 m is dug to a depth of 12 km. Earth taken out of it has been evenly spread all around it to a width of 7 m to form an embankment. Find the height of the embankment so formed.

Answer 1

Hint: We will first calculate the volume of earth/soil that has been dug out to form the well. We will convert the depth of the well from km to m, since all other measurements are given in m. The shape of the embankment is a thick cylinder with inner radius the same as the radius of the well and the outer radius is the radius of the well added to the width of the embankment. We will use the formula for the volume of the embankment to find its height.

Complete step by step answer:
The diameter of the well is 14 m. Hence, the radius is 7 m. The inner radius of the well and the embankment is the same. Let this radius be denoted by $r$ and its value is $r=7\text{ m}$. The depth of the well is 12 km. Since all other measurements are given in m, we will convert the depth of the well from km to m. So we have a depth of 12,000 m.
The well has a cylinder shape with radius 7 m and height 12000 m. The formula for calculating the volume of a cylinder is
${{V}_{cylinder}}=\pi {{r}^{2}}h$
Substituting the values of $r=7\text{ m}$ and $h=12000\text{ m}$, we get
${{V}_{cylinder}}=\pi \times {{\left( 7 \right)}^{2}}\times 12000$
The volume of the earth/soil taken out is equal to the volume of the well. Therefore, from the above equation, we get
\[{{V}_{cylinder}}={{V}_{soil}}=588000\pi \]
Now, we know that the soil dug out is used to form the embankment. Therefore, the volume of the embankment is the same as the volume of the soil. The embankment is a thick cylinder, with inner radius $r=7\text{ m}$ and outer radius $r+w=7+7=14\text{ m}$, where $w$ is the width of the embankment.
The formula for calculating the volume of the embankment will be,
${{V}_{embankment}}={{V}_{outer}}-{{V}_{inner}}$
where ${{V}_{outer}}$ is the volume of the cylinder formed with the outer radius and ${{V}_{inner}}$ is the volume of the inner cylinder. Also, we have ${{V}_{embankment}}={{V}_{soiil}}$. Substituting these values in the above formula, we get
$588000\pi =\pi {{\left( r+w \right)}^{2}}{{h}_{e}}-\pi {{r}^{2}}{{h}_{e}}$
where ${{h}_{e}}$ is the height of the embankment. Substituting the inner and outer radii of the embankment, we get
$588000\pi =\pi {{\left( 14 \right)}^{2}}{{h}_{e}}-\pi {{\left( 7 \right)}^{2}}{{h}_{e}}$
Simplifying the above equation and solving for ${{h}_{e}}$, we get
\[\begin{align}
  & ~588000=196{{h}_{e}}-49{{h}_{e}} \\
 & \therefore 588000=147{{h}_{e}} \\
 & \therefore {{h}_{e}}=\dfrac{588000}{147} \\
 & \therefore {{h}_{e}}=4000\text{ m} \\
\end{align}\]
Therefore, the height of the embankment is 4000 m, that is 4 km.

Note:
The important part in this question is the relation between volumes of different objects. It is essential to write down the formulae for the volumes of different objects explicitly, since it is possible to misplace the values of variables. Visualizing the different objects mentioned in the question is useful.