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.

Answer

To solve the problem step by step, we need to find the height of the embankment formed by the earth dug out from the well.

### Step 1: Calculate the volume of the earth dug out from the well.
The volume \( V \) of a cylinder (which is the shape of the well) is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or depth in this case).

- The diameter of the well is 3 m, so the radius \( r \) is:
\[ r = \frac{3}{2} = 1.5 \text{ m} \]
- The depth \( h \) of the well is 14 m.

Now substituting the values into the volume formula:
\[ V = \pi (1.5)^2 (14) \]
\[ = \pi (2.25) (14) \]
\[ = 31.5\pi \text{ m}^3 \]

### Step 2: Calculate the area of the embankment.
The embankment is in the shape of a circular ring. To find the area of the embankment, we need to calculate the area of the larger circle (outer radius) and subtract the area of the smaller circle (inner radius).

- The outer radius \( R \) of the embankment is:
\[ R = r + \text{width} = 1.5 + 4 = 5.5 \text{ m} \]
- The inner radius \( r \) is:
\[ r = 1.5 \text{ m} \]

Now, we calculate the area of the larger circle and the smaller circle:
\[ \text{Area of the larger circle} = \pi R^2 = \pi (5.5)^2 = \pi (30.25) \]
\[ \text{Area of the smaller circle} = \pi r^2 = \pi (1.5)^2 = \pi (2.25) \]

Now, the area of the embankment \( A \) is:
\[ A = \text{Area of the larger circle} - \text{Area of the smaller circle} \]
\[ = \pi (30.25 - 2.25) = \pi (28) \]

### Step 3: Determine the height of the embankment.
Let \( h \) be the height of the embankment. The volume of the earth dug out is equal to the volume of the embankment:
\[ \text{Volume of the embankment} = \text{Area of the embankment} \times \text{height of the embankment} \]
\[ 31.5\pi = \pi (28) h \]

Dividing both sides by \( \pi \):
\[ 31.5 = 28h \]

Now, solving for \( h \):
\[ h = \frac{31.5}{28} = 1.125 \text{ m} \]

### Final Answer:
The height of the embankment is \( 1.125 \text{ m} \).

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