Question : A well with an inner radius of 3 m, is dug 6 m deep. The soil taken out of it has been spread evenly all around it to a width of 2 m to form an embankment. The height (in m) of the embankment is:
Option 1: $4 \frac{1}{2}$
Option 2: $4 \frac{1}{4}$
Option 3: $3 \frac{1}{4}$
Option 4: $3 \frac{3}{8}$
Correct Answer: $3 \frac{3}{8}$
Solution :
The volume of the well (a cylinder) is $\pi r^2 h$, where $r$ is the radius and $h$ is the height.
Substituting r = 3 m and h = 6 m.
The volume of the well = $\pi r^2 h= \pi \times 3^2 \times 6 =54\pi \;m^3$
The embankment is in the shape of a hollow cylinder with an outer radius, R = r + width = 3 + 2 = 5 m and an inner radius r = 3 m.
Let the height of the embankment be $h_2$.
The volume of the embankment = $\pi (R^2 - r^2) h_2$
Equating the volume of the well and the embankment,
$⇒54\pi = \pi (5^2 - 3^2) h_2$
Solving for $h_2$,
$⇒h_2 = \frac{54}{16} = \frac{27}{8} = 3\frac{3}{8}\;m$
Hence, the correct answer is $3\frac{3}{8}$.
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