Question

A well with inner diameter 14m is 15m deep and the earth taken out is spread all around to a width of 7m to form an embankment. Find the height of the embankment.

Answer 1

Hint: We solve this question by first finding the radius of the well from the given diameter. Then we measured the volume of the earth dug using the formula for volume for cylinder, $Volume=\pi {{r}^{2}}h$. Then we find the outer radius of the well. Then we find the area of the embankment using the formula for the area of the circle, $Area=\pi {{r}^{2}}$. Then we assume the height of the embankment as H and find the volume of the embankment by multiplying the area with height and then equate it with the volume of the earth dug out and solve it to find the value of H, that is height of the embankment.

Complete step by step answer:
Here we are given that a well is dug with inner diameter 14m and is of depth 15m.
So, we can understand that the well is a cylinder with diameter 14m and height 15m.
Then radius of the cylinder is,
$\begin{align}
  & \Rightarrow radius=\dfrac{diameter}{2} \\
 & \Rightarrow radius=\dfrac{14}{2}=7 \\
\end{align}$
So, the radius of the well is 7m and height 15m.

We are also given that the earth dug is spread around the well to form an embankment.
So, first let us find the volume of the earth taken out.
Now let us consider the formula for the volume of the cylinder.
$Volume=\pi {{r}^{2}}h$
So, using it we get the volume of earth dug out as,
\[\Rightarrow Volume\ dug\ out=\pi \times {{\left( 7 \right)}^{2}}\times 15\]
Let us take the value of $\pi $ as $\dfrac{22}{7}$. Then we get the volume as,
\[\begin{align}
  & \Rightarrow Volume\ dug\ out=\dfrac{22}{7}\times {{\left( 7 \right)}^{2}}\times 15 \\
 & \Rightarrow Volume\ dug\ out=22\times 7\times 15 \\
 & \Rightarrow Volume\ dug\ out=2310 \\
\end{align}\]
So, we get the volume of the earth dug as 2310 cubic meters.
We are given that the earth dug out is spread all around to a width of 7m to form an embankment.

So, because of the embankment the outer radius of the well is,
$\begin{align}
  & \Rightarrow Outer\ Radius=Inner\ Radius+Width\ of\ Embankment \\
 & \Rightarrow Outer\ Radius=7+7 \\
 & \Rightarrow Outer\ Radius=14m \\
\end{align}$
Now let us find the area of the embankment. As it is in the form of a circle, let us consider the formula for the area of a circle.
$Area=\pi {{r}^{2}}$
So, using this formula we get the area of embankment as,
$\Rightarrow Area\text{ }of\ embankment=\pi {{\left( Outer\ Radius \right)}^{2}}-\pi {{\left( Inner\ Radius \right)}^{2}}$
Substituting the above values of outer radius and inner radius, we get,
 \[\begin{align}
  & \Rightarrow Area\text{ }of\ embankment=\pi {{\left( 14 \right)}^{2}}-\pi {{\left( 7 \right)}^{2}} \\
 & \Rightarrow Area\text{ }of\ embankment=196\pi -49\pi \\
 & \Rightarrow Area\text{ }of\ embankment=\left( 196-49 \right)\pi \\
 & \Rightarrow Area\text{ }of\ embankment=147\pi \\
\end{align}\]
Let us take the value of $\pi $ as $\dfrac{22}{7}$. Then we get the area as,
\[\begin{align}
  & \Rightarrow Area\text{ }of\ embankment=\dfrac{22}{7}\times 147 \\
 & \Rightarrow Area\text{ }of\ embankment=22\times 21 \\
 & \Rightarrow Area\text{ }of\ embankment=462 \\
\end{align}\]
So, we get the area of embankment as 462 sq. m.
Now let us find the volume of the embankment.
Now let us assume that the height of the embankment is $H$.
$\begin{align}
  & \Rightarrow Volume\ of\ embankment=\left( Area\ of\ embankment \right)\times \left( Height \right) \\
 & \Rightarrow Volume\ of\ embankment=\left( 462 \right)\times \left( H \right) \\
 & \Rightarrow Volume\ of\ embankment=462H \\
\end{align}$
As the volume of the earth dug out is used to make an embankment, volume of embankment is equal to the volume of earth taken out.
So, let us equate the values from equations (1) and (2). Then we get,
$\begin{align}
  & \Rightarrow 462H=2310 \\
 & \Rightarrow H=\dfrac{2310}{462} \\
 & \Rightarrow H=5m \\
\end{align}$
So, we get the value of the height of the embankment as 5m.

So, the correct answer is “5m”.

Note: There is a possibility of one making a mistake while solving this problem by finding the Area of the embankment without subtracting the area of inner circle, that is, by finding the area of embankment as,
$\Rightarrow Area\text{ }of\ embankment=\pi {{\left( Outer\ Radius \right)}^{2}}$
But it is wrong as the embankment is not the complete outer circle, it is the area of the outer circle minus the area of inner circle.