A well of diameter 3 m is dug 14 m deep. The soil taken out of it is spread evenly all around it to a width of 5 m to form an embankment. Find the height of the embankent.

Answer

To solve the problem step by step, we will calculate the height of the embankment formed by the soil dug out from the well.

### Step 1: Calculate the Volume of the Well
The well is in the shape of a cylinder. 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 of the cylinder

**Given:**
- Diameter of the well = 3 m, so the radius \( r = \frac{3}{2} = 1.5 \) m
- Depth of the well \( h = 14 \) m

**Substituting the values:**

\[ V = \pi (1.5)^2 (14) \]

Calculating \( (1.5)^2 \):

\[ (1.5)^2 = 2.25 \]

Now substituting back into the volume formula:

\[ V = \pi \times 2.25 \times 14 \]

Calculating \( 2.25 \times 14 \):

\[ 2.25 \times 14 = 31.5 \]

Thus, the volume of the well is:

\[ V = 31.5\pi \, \text{m}^3 \]

### Step 2: Calculate the Area of the Embankment
The embankment is formed around the well and has a width of 5 m. The outer radius \( R \) of the embankment is:

\[ R = r + 5 = 1.5 + 5 = 6.5 \, \text{m} \]

The area \( A \) of the embankment (ring) can be calculated using the formula for the area of a circular ring:

\[ A = \pi (R^2 - r^2) \]

Substituting the values:

\[ A = \pi ((6.5)^2 - (1.5)^2) \]

Calculating \( (6.5)^2 \) and \( (1.5)^2 \):

\[ (6.5)^2 = 42.25 \]
\[ (1.5)^2 = 2.25 \]

Now substituting back into the area formula:

\[ A = \pi (42.25 - 2.25) = \pi (40) \]

Thus, the area of the embankment is:

\[ A = 40\pi \, \text{m}^2 \]

### Step 3: Calculate the Height of the Embankment
The height \( h_e \) of the embankment can be found using the formula:

\[ h_e = \frac{\text{Volume of soil}}{\text{Area of embankment}} \]

Substituting the values we calculated:

\[ h_e = \frac{31.5\pi}{40\pi} \]

The \( \pi \) cancels out:

\[ h_e = \frac{31.5}{40} \]

Calculating the height:

\[ h_e = 0.7875 \, \text{m} \]

### Conclusion
The height of the embankment is approximately:

\[ h_e \approx 0.79 \, \text{m} \]