Question

A well of diameter $4m$ is dug $21m$ deep. The earth taken out of it has been spread evenly all around it in the shape of a ring of width $3m$ to form an embankment. Find the height of the embankment.

Answer 1

Hint: Proceed the solution by using the condition that volume of earth is equal to volume of embankment. We know that volume of embankment $V = \pi ({R^2} - {r^2})h$ and volume of earth is $(v) = \pi {r^2}d$.

Here let us solve solution by gathering the data given to us, so here the
Diameter of well = $4m$
Radius of well $(r)$ = $2cm$ [Radius is half of the diameter]
Depth of earth $(d)$ = $21m$
We know that volume of earth as $(v) = \pi {r^2}d$
$ \Rightarrow v = \dfrac{{22}}{7} \times 2 \times 2 \times 21$
$ \Rightarrow v = 264{m^3}$
Given the width of embankment = $3m$
Outer radius of ring $(R)$ = $2 + 3 = 5m$
Let the height of embankment $h$
We know that volume of embankment $V = \pi ({R^2} - {r^2})h$
Here,
 Volume of embankment = Volume of earth
$ \Rightarrow $$\
  \pi ({R^2} - {r^2})h = \pi {r^2}d \\
    \\
\ $
On substituting the values we get
$\
   \Rightarrow \dfrac{{22}}{7} \times (25 - 4) \times h = 264 \\
   \Rightarrow h = \dfrac{{264 \times 7}}{{22 \times 21}} \\
   \Rightarrow h = 4 m \\
\ $
Therefore the height of embankment =$4m$
NOTE: Outer ring radius has to be calculated by using the inner radius which is used in the volume of embankment to find height.