A well with 10 m inside diameter is dug 8.4 m deep. Earth taken out of it is spread all around it to a width of 7.5 m to form an embankment. Find the height of the embankment.

Radius of the well, r = $\dfrac{\text{diameter}}{2} = \dfrac{10}{2} = 5$ m

Depth of the well (h) = 8.4 m

Volume of the earth dug out from the well = πr²h

$= \dfrac{22}{7} \times 5^2 \times 8.4 \\[1em] = \dfrac{22}{7} \times 25 \times 8.4 \\[1em] = \dfrac{4620}{7} \\[1em] = 660 \text{ m}^3$

External radius, R = radius of the well + 7.5 = 5 + 7.5 = 12.5 m

The embankment forms a hollow cylinder around the well.

Let height of the embankment be H.

∴ Volume of embankment = πR²H - πr²H

= πH(R² - r²)

$= \dfrac{22}{7} \times \text{H} \times [(12.5)^2 - 5^2] \\[1em] = \dfrac{22}{7} \times \text{H} \times (156.25 - 25) \\[1em] = \dfrac{22}{7} \times \text{H} \times 131.25 \\[1em] = \dfrac{2887.5}{7} \times \text{H} \\[1em] = 412.5 \text{ H} \text{ m}^3$

Volume of earth dug out = Volume of the embankment

$\Rightarrow 660 = 412.5 \text{H} \\[1em] \Rightarrow \text{H} = \dfrac{660}{412.5} \\[1em] \Rightarrow \text{H} = 1.6 \text{ m.}$

Hence, the height of the embankment is 1.6 m.