A well of diameter 4 m is dug 21 m deep . The earth taken out of ot has been spread envenly all around it in the shape of a circular ring of width 3 m to form an embankment .Find the height of the embankment .

Answer

To solve the problem, we need to find the height of the embankment formed by the earth taken out of the well. Let's break it down step by step.

### Step 1: Calculate the volume of the earth taken 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 4 m, so the radius \( r \) is:
\[ r = \frac{4}{2} = 2 \text{ m} \]
- The depth of the well \( h \) is 21 m.

Now we can calculate the volume:
\[ V = \pi (2)^2 (21) = \pi \times 4 \times 21 = 84\pi \text{ cubic meters} \]

### Step 2: Determine the dimensions of the circular ring (embankment).
The embankment is in the shape of a circular ring with a width of 3 m.

- The inner radius (the radius of the well) is 2 m.
- The outer radius \( R \) of the embankment is:
\[ R = 2 + 3 = 5 \text{ m} \]

### Step 3: Calculate the area of the circular ring.
The area \( A \) of the circular ring can be calculated by finding the area of the larger circle and subtracting the area of the smaller circle:
\[ A = \pi R^2 - \pi r^2 \]
Substituting the values:
\[ A = \pi (5^2) - \pi (2^2) = \pi (25 - 4) = 21\pi \text{ square meters} \]

### Step 4: Find the height of the embankment.
Let the height of the embankment be \( h_e \). The volume of the embankment can also be expressed as:
\[ \text{Volume of embankment} = \text{Area of ring} \times \text{Height of embankment} \]
We know the volume of the earth taken out from the well is equal to the volume of the embankment:
\[ 84\pi = 21\pi \times h_e \]
Dividing both sides by \( 21\pi \):
\[ h_e = \frac{84\pi}{21\pi} = 4 \text{ m} \]

### Final Answer:
The height of the embankment is **4 meters**.

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