Question

A well 20m in diameter is dug 14m deep and the earth taken out is spread all round it to a width of 5m to form an embankment. The height of the embankment is
A) 10m
B) 11m
C) 11.2m
D) 11.5m

Answer 1

Hint: This problem is based on the concept of cylinder. We solve this question with the help of formula i.e. volume of cylinder.
∴Volume of cylinder = $\pi {r^2}h$
Where, r = radius of cylinder
and h= height of cylinder
The embankment would form a hollow cylinder with height (H) and radius (R) equals to the sum of radius of well (r) and the width of embankment.
∴ Volume of hollow cylinder = $\pi \left( {{R^2} - {r^2}} \right)H$
Now, we can find the height by equating both the volumes.

Complete step-by-step answer:
Given that,
Diameter of well = 20m
∴ Radius of well (r) = $\dfrac{{{\text{Diameter}}}}{2} = \dfrac{{20}}{2}$
r = 10m
Height of well (h) = 14m
Width of embankment = 5m
The earth taken out is spread all around the well, therefore, the radius of embankment if taken from the centre will be R.

$\begin{gathered}
  R = {\text{Radius of well + width}} \\
  R = r + 5 \\
  R = 10 + 5 \\
  R = 15m \\
\end{gathered} $
Now, to find the height of embankment we have to equate the volume of well and volume of embankment.
Volume of well = Volume of embankment
$\begin{gathered}
  \pi {r^2}h = \pi \left( {{R^2} - {r^2}} \right) \times H \\
  \pi {\left( {10} \right)^2} \times 14 = \pi \left( {{{15}^2} - {{10}^2}} \right) \times H \\
  \pi {\left( {10} \right)^2} \times 14 = \pi \left( {225 - 100} \right) \times H \\
  \pi {\left( {10} \right)^2} \times 14 = \pi \left( {125} \right) \times H \\
  H = \dfrac{{100 \times 14}}{{125}} \\
  H = \dfrac{{56}}{5} \\
  H = 11.2m \\
\end{gathered} $
∴ The height of the embankment = 11.2m

∴ Option (C) is correct.

Note: It can also be solved with the concept of circular ring. The embankment would form a circular ring with internal radius same as the radius of the well (r) and external radius (R) equals to the sum of radius of well and the width of embankment. By using, the formula of area of circular ring,
∴ Area of circular ring (embankment) = $\pi \left( {{R^2} - {r^2}} \right)$
and Volume of well = $\pi {r^2}h$
Height of embankment = $\dfrac{{{\text{Volume of well}}}}{{{\text{Area of embankment}}}}$