# 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.

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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.

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.

Last updated date: 29th May 2023

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