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
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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.
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