A well with 12 m inside diameter is dug 21 m deep. Earth taken out of it is spread all around to a width of 6 m to form an embankment. Find the height of the embankment.

Options

A. 6 m
B. 5 m
C. 8 m
D. 7 m

Answer

To solve the problem step-by-step, we need to calculate the height of the embankment formed by the earth taken out from the well.

### Step 1: Calculate the volume of the well

The well is cylindrical in shape. The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the cylinder
- \( h \) is the height (or depth in this case)

Given:
- Diameter of the well = 12 m, hence the radius \( r = \frac{12}{2} = 6 \) m
- Depth of the well \( h = 21 \) m

Now, substituting the values into the volume formula:

\[ V = \pi (6)^2 (21) \]
\[ V = \pi (36)(21) \]
\[ V = 756\pi \, \text{m}^3 \]

### Step 2: Calculate the volume of the embankment

The embankment is formed around the well and is also cylindrical. The outer radius of the embankment is the radius of the well plus the width of the embankment.

- Width of the embankment = 6 m
- Outer radius \( R = 6 + 6 = 12 \) m

The height of the embankment is \( h \) (which we need to find).

The volume of the embankment can be calculated as the difference between the volume of the larger cylinder (outer radius) and the smaller cylinder (inner radius):

\[ V_{\text{embankment}} = V_{\text{outer}} - V_{\text{inner}} \]

Calculating the volume of the outer cylinder:

\[ V_{\text{outer}} = \pi R^2 h = \pi (12)^2 h = 144\pi h \]

Calculating the volume of the inner cylinder (the well):

\[ V_{\text{inner}} = \pi r^2 h = \pi (6)^2 h = 36\pi h \]

Thus, the volume of the embankment becomes:

\[ V_{\text{embankment}} = 144\pi h - 36\pi h = 108\pi h \]

### Step 3: Set the volumes equal

Since the volume of the earth taken out from the well is equal to the volume of the embankment, we can set the two volumes equal:

\[ 756\pi = 108\pi h \]

### Step 4: Solve for \( h \)

Dividing both sides by \( \pi \):

\[ 756 = 108h \]

Now, solving for \( h \):

\[ h = \frac{756}{108} \]
\[ h = 7 \, \text{m} \]

### Final Answer

The height of the embankment is \( 7 \, \text{m} \).

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