A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.

Explanation

Given the height of deep well which form a cylinder `(h_(1))` =14m
and radius of deep well `(r_(1))=(3)/(2)`
Let the width and height of the embankement form a circular ring are d and `h_(2)`
Here,
Now the voluem of deep well (cylinder)
`=pir_(1)^(2)h_(1)=(22)/(7)xx(3)/(2)xx(3)/(2)xx14`
=99 `m^(3)`
Here, the embankemnt form a hollow cylinder
`therefore` The volume of water filled in a spherical galss vessel
= volume of cylinder+ volume of sphere
`=pir_(1)^(2)h_(1)+(4)/(3)pir_(2)^(3)`
Let the width and height of the embankment form a circular ring are d and `h_(2)`
Here,
Now the volume of deep well (cylinder)
`=pir_(1^(2))h+_(1)=(22)/(7)xx(3)/(2)xx(3)/(2)xx(3)/(2)xx14`
=`99m^(3)`
Here the embankemnet form a hollow cylinder
`therefore` Volume of hollow cylinder (embankment) =`pi(r_(1)+d)^(2).h_(2)-pir_(1)^(2)h_(2)^(2)`
`=pi(3)/(2+4^(2))xxh_(2)0pi((3)/(2^(2)))h_(2)=pi[((11)/(2^(2)))h_(2)-(9)/(4)h_(2)]`
Now accordinng of question,
volume of deep well (cylinder)=volume of embankenet (hollow cylinder)
`99=pi[(11)/(2)^(2)xxh_(2)-(9)/((4)h_(2))]`
`=(7xx99)/(22)=(121)/(4)xxh_(2)-(9)/(4)h_(2)=(112 h_(2))/(4)`
`h_(2)=(7xx99xx4)/(22xx112)=(9)/(8)=11.25`