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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.
Given:
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.
To do:
We have to find the height of the embankment.
Solution:
Diameter of the well $=3\ m$
This implies,
Inner radius of the well $r=\frac{3}{2}=1.5 \mathrm{~m}$
Depth of the well $h=14 \mathrm{~m}$
This implies,
Volume of the earth dug out $=\pi r^{2} h$
$=\frac{22}{7} \times 1.5 \times 1.5 \times 14$
$=99 \mathrm{~m}^{3}$ Width of the embankment $w=4 \mathrm{~m}$
This implies,
Outer radius $\mathrm{R}=1.5+4$
$=5.5 \mathrm{~m}$
Let $h$ be the height of the embankment.
Therefore,
$\pi(\mathrm{R}^{2}-r^{2}) h=99$
$\Rightarrow \frac{22}{7}[(5.5)^{2}-(1.5)^{2}] h=99$
$\Rightarrow \frac{22}{7}[30.25-2.25] h=99$
$\Rightarrow \frac{22}{7} \times 28 h=99$
$\Rightarrow h=\frac{99 \times 7}{22 \times 28}$
$\Rightarrow h=\frac{9 \times 1}{2\times4}$
$\Rightarrow h=\frac{9}{8}\ m$
The height of the embankment is $\frac{9}{8} \mathrm{~m}$.