The diameter of a hollow cylindrical vessel is 14 cm .There is some water in it. When a cubical iron piece is fully immeresed in it , the water surface rises by `8(9)/(14)`cm .Find the edge of the cube.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have a hollow cylindrical vessel with a diameter of 14 cm. When a cubical iron piece is fully immersed in it, the water level rises by \(8 \frac{9}{14}\) cm. We need to find the edge length of the cube.

### Step 2: Calculate the radius of the cylinder
The radius \( r \) of the cylinder can be calculated from the diameter:
\[ r = \frac{\text{diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \]

### Step 3: Convert the rise in water level to an improper fraction
The rise in water level is given as \(8 \frac{9}{14}\) cm. We convert this mixed fraction to an improper fraction:
\[ 8 \frac{9}{14} = \frac{8 \times 14 + 9}{14} = \frac{112 + 9}{14} = \frac{121}{14} \text{ cm} \]

### Step 4: Calculate the volume of water displaced
The volume of water displaced when the cube is immersed is equal to the volume of the cylinder that corresponds to the rise in water level. The volume \( V \) of the cylindrical section can be calculated using the formula:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V = \pi \times (7 \text{ cm})^2 \times \frac{121}{14} \text{ cm} \]
\[ V = \pi \times 49 \text{ cm}^2 \times \frac{121}{14} \text{ cm} \]

### Step 5: Substitute the value of \(\pi\)
Using \(\pi \approx \frac{22}{7}\):
\[ V = \frac{22}{7} \times 49 \times \frac{121}{14} \]

### Step 6: Simplify the expression
First, simplify \( \frac{22 \times 49 \times 121}{7 \times 14} \):
\[ = \frac{22 \times 49 \times 121}{98} \]
Calculating \( 22 \times 49 = 1078 \):
\[ = \frac{1078 \times 121}{98} \]
Now, simplifying \( \frac{1078 \times 121}{98} \):
\[ = \frac{130678}{98} \]

### Step 7: Calculate the volume of the cube
The volume of the cube is given by \( a^3 \), where \( a \) is the edge length of the cube. Thus, we have:
\[ a^3 = \frac{130678}{98} \]

### Step 8: Find the edge length of the cube
To find \( a \), we need to take the cube root of the volume:
\[ a = \sqrt[3]{\frac{130678}{98}} \]
Calculating this gives:
\[ a = 11 \text{ cm} \]

### Conclusion
The edge of the cube is \( 11 \text{ cm} \).

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