A well 20 m in diameter is dug 14m deep and the earth taken out is spread all around it to a width of 5 m to form an embankment. The height of the embankment is :

Options

A. 10 m
B. 11 m
C. 11.2 m
D. 11.5 m

Answer

To find the height of the embankment formed by the earth removed from the well, we can follow these steps:

### Step 1: Calculate the Volume of the Well
The well is cylindrical in shape. The formula for the volume 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 = 20 m, therefore the radius \( r = \frac{20}{2} = 10 \) m.
- Depth of the well (height) \( h = 14 \) m.

Now, substituting the values into the formula:
\[ V_{\text{well}} = \pi (10^2)(14) = \pi (100)(14) = 1400\pi \, \text{m}^3 \]

### Step 2: Determine the Dimensions of the Embankment
The embankment is formed around the well with a width of 5 m.

- Inner radius of the embankment (which is the radius of the well) \( r_1 = 10 \) m.
- Outer radius of the embankment \( r_2 = r_1 + 5 = 10 + 5 = 15 \) m.

### Step 3: Calculate the Volume of the Embankment
The volume of the embankment can be found by subtracting the volume of the inner cylinder (the well) from the volume of the outer cylinder (the embankment).

The volume of the outer cylinder is:
\[ V_{\text{outer}} = \pi r_2^2 H \]
Where \( H \) is the height of the embankment we want to find.

The volume of the inner cylinder (the well) is:
\[ V_{\text{inner}} = \pi r_1^2 h = 1400\pi \, \text{m}^3 \]

Thus, the volume of the embankment is:
\[ V_{\text{embankment}} = V_{\text{outer}} - V_{\text{inner}} = \pi r_2^2 H - 1400\pi \]

Substituting \( r_2 = 15 \):
\[ V_{\text{embankment}} = \pi (15^2) H - 1400\pi \]
\[ = \pi (225) H - 1400\pi \]

### Step 4: Set the Volumes Equal
Since the volume of the earth removed from the well is equal to the volume of the embankment:
\[ 1400\pi = \pi (225) H - 1400\pi \]
Dividing through by \( \pi \):
\[ 1400 = 225H - 1400 \]
Adding \( 1400 \) to both sides:
\[ 2800 = 225H \]
Now, solving for \( H \):
\[ H = \frac{2800}{225} \approx 12.44 \, \text{m} \]

### Step 5: Final Calculation
To find the height \( H \):
\[ H \approx 12.44 \, \text{m} \]

However, we need to ensure the correct calculation. Let's recalculate:
\[ H = \frac{1400}{225 - 100} = \frac{1400}{125} = 11.2 \, \text{m} \]

### Conclusion
The height of the embankment is \( 11.2 \) m.