A cylindrical tank of radius 5.6 m and depth of 'h' m is built by digging out earth. The sand taken out is spread all around the tank to form a circular embankment to a width of 7m. What is the depth of the tank if the height of the embankment is 1.97m ?

Options

A. 4.2 m
B. 7 m
C. 8 m
D. 6.7 m

Answer

To solve the problem step by step, we will find the depth of the cylindrical tank using the volumes of the tank and the embankment.

### Step 1: Identify the dimensions of the tank and embankment
- Radius of the cylindrical tank (r) = 5.6 m
- Width of the embankment = 7 m
- Height of the embankment = 1.97 m
- Depth of the tank (h) = ?

### Step 2: Calculate the outer radius of the embankment
The outer radius (R) of the embankment is the sum of the radius of the tank and the width of the embankment:
\[ R = r + \text{width of embankment} = 5.6 \, \text{m} + 7 \, \text{m} = 12.6 \, \text{m} \]

### Step 3: Calculate the volume of the cylindrical tank
The volume (V_tank) of the cylindrical tank can be calculated using the formula:
\[ V_{tank} = \pi r^2 h \]
Substituting the known values:
\[ V_{tank} = \pi (5.6)^2 h \]

### Step 4: Calculate the volume of the circular embankment
The volume (V_embankment) of the circular embankment can be calculated using the formula:
\[ V_{embankment} = \pi (R^2 - r^2) \times \text{height of embankment} \]
Substituting the known values:
\[ V_{embankment} = \pi ((12.6)^2 - (5.6)^2) \times 1.97 \]

### Step 5: Set the volumes equal to each other
Since the volume of the sand taken out from the tank is equal to the volume of the embankment formed:
\[ V_{tank} = V_{embankment} \]
So we have:
\[ \pi (5.6)^2 h = \pi ((12.6)^2 - (5.6)^2) \times 1.97 \]

### Step 6: Simplify the equation
We can cancel out π from both sides:
\[ (5.6)^2 h = ((12.6)^2 - (5.6)^2) \times 1.97 \]

### Step 7: Calculate the values
Now we calculate the squares:
- \( (5.6)^2 = 31.36 \)
- \( (12.6)^2 = 158.76 \)

Now substituting these values:
\[ 31.36 h = (158.76 - 31.36) \times 1.97 \]
\[ 31.36 h = 127.4 \times 1.97 \]
\[ 31.36 h = 250.058 \]

### Step 8: Solve for h
Now we can solve for h:
\[ h = \frac{250.058}{31.36} \]
Calculating this gives:
\[ h \approx 7.98 \, \text{m} \]

### Step 9: Round to the nearest whole number
Rounding 7.98 to the nearest whole number gives:
\[ h \approx 8 \, \text{m} \]

### Conclusion
The depth of the tank is approximately **8 meters**.