A well of diameter 3m 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 4m to form an embankment. Find the height of the embankment.

Options

A. 4.25 m
B. 2.25 m
C. 1.125m
D. 1.75 m

Answer

To solve the problem step by step, we will first calculate the volume of the earth that was dug out from the well and then use that volume to find the height of the embankment formed by spreading the earth around 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 = 3 m, therefore, the radius \( r = \frac{3}{2} = 1.5 \) m
- Depth of the well \( h = 14 \) m

Now, substituting the values into the volume formula:
\[ V = \pi (1.5)^2 (14) \]
\[ V = \pi (2.25) (14) \]
\[ V = \pi (31.5) \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 31.5 = \frac{693}{7} = 99 \text{ m}^3 \]

### Step 2: Calculate the dimensions of the embankment
The embankment is in the shape of a circular ring. The width of the embankment is given as 4 m.

The inner radius of the embankment (which is the radius of the well) is:
\[ r_1 = 1.5 \text{ m} \]
The outer radius of the embankment is:
\[ r_2 = r_1 + \text{width} = 1.5 + 4 = 5.5 \text{ m} \]

### Step 3: Calculate the volume of the embankment
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).

The volume of the embankment \( V_{embankment} \) is given by:
\[ V_{embankment} = \pi (r_2^2 - r_1^2) h \]
Where \( h \) is the height of the embankment.

Substituting the values:
\[ V_{embankment} = \pi ((5.5)^2 - (1.5)^2) h \]
Calculating the squares:
\[ (5.5)^2 = 30.25 \quad \text{and} \quad (1.5)^2 = 2.25 \]
Thus,
\[ V_{embankment} = \pi (30.25 - 2.25) h = \pi (28) h \]

### Step 4: Equate the volumes
Since the volume of the earth dug out from the well is equal to the volume of the embankment:
\[ 99 = \pi (28) h \]

Substituting \( \pi \approx \frac{22}{7} \):
\[ 99 = \frac{22}{7} (28) h \]
\[ 99 = \frac{616}{7} h \]

### Step 5: Solve for \( h \)
To find \( h \), we rearrange the equation:
\[ h = \frac{99 \times 7}{616} \]
Calculating:
\[ h = \frac{693}{616} \approx 1.125 \text{ m} \]

### Final Answer
The height of the embankment is approximately \( 1.125 \) m.