Question

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 circular ring of width 4m to form an embankment. Find the height of the embankment.

Answer 1

Both well and embankment are in the form of cylinder. Let well be cylinder A and embankment be cylinder B.

$\because$ mud of well is distributed in embankment

Volume of well $=$ Volume of embankment

Volume of well -

For A, $r=\frac{3}{2}m=1.5m,height=14m$

$\therefore$ Volume of cylinder A $=\pi r^{2}h$

$=\pi \times {\left(1.5\right)}^2\times 14$

$=31.5\pi m^3$

$\therefore$ Volume of well $=$ Volume of cylinder A $=31.5\pi m^3$

Volume of embankment -

For cylinder B,

Cylinder B is a hollow cylinder with inner radius $r_1=\frac{3}{2}=1.5m$

$\Rightarrow$ External radius $r_2=$ internal radius + width

$=1.5+4$

$=5.5m$

$\Rightarrow$ Volume of cylinder with internal radius $=\pi r_1^{2}h$

$=\pi h{\left(1.5\right)}^2$

$\Rightarrow$ Volume of cylinder with external radius $=\pi r_2^{2}h$

$=\pi h{\left(5.5\right)}^2$

$\therefore$ Volume of cylinder B $=\pi h{\left(5.5\right)}^2-\pi h{\left(1.5\right)}^2$

$=\pi h(30.25-2.25)$

$=\pi h\left(28\right)$

$=28\pi h{m}^3$

$\Rightarrow$ Volume of embankment $=28\pi h{m}^3$

Now, volume of well $=$ volume of embankment

$\Rightarrow 31.5\pi =28\pi h$

$\therefore h=\frac{31.5}{28}=1.125m.$

Hence, the answer is $1.125m.$