A rectangular vessel 22 cm by 16 cm by 14 cm is full of water. If the total water is poured into the empty cylindrical vessel of radius 8 cm, find the height of water in the cylinderical vessel.

Answer

To solve the problem step by step, we will find the height of water in the cylindrical vessel after pouring the water from the rectangular vessel.

### Step 1: Calculate the volume of the rectangular vessel.
The volume \( V \) of a rectangular vessel is given by the formula:
\[ V = \text{Length} \times \text{Breadth} \times \text{Height} \]
Given:
- Length \( L = 22 \) cm
- Breadth \( B = 16 \) cm
- Height \( H = 14 \) cm

Substituting the values:
\[ V = 22 \, \text{cm} \times 16 \, \text{cm} \times 14 \, \text{cm} \]
Calculating this:
\[ V = 22 \times 16 = 352 \, \text{cm}^2 \]
\[ V = 352 \times 14 = 4928 \, \text{cm}^3 \]

### Step 2: Set up the volume of the cylindrical vessel.
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi R^2 H \]
Where:
- \( R \) is the radius of the cylinder
- \( H \) is the height of the water in the cylinder

Given:
- Radius \( R = 8 \) cm

### Step 3: Equate the volumes.
Since all the water from the rectangular vessel is poured into the cylindrical vessel, the volumes are equal:
\[ 4928 \, \text{cm}^3 = \pi \times (8 \, \text{cm})^2 \times H \]

### Step 4: Substitute the value of \( \pi \).
Using \( \pi \approx \frac{22}{7} \):
\[ 4928 = \frac{22}{7} \times 64 \times H \]
Calculating \( 64 \times \frac{22}{7} \):
\[ 64 \times \frac{22}{7} = \frac{1408}{7} \]
So, we have:
\[ 4928 = \frac{1408}{7} \times H \]

### Step 5: Solve for \( H \).
To isolate \( H \), multiply both sides by \( 7 \):
\[ 4928 \times 7 = 1408 \times H \]
Calculating \( 4928 \times 7 \):
\[ 4928 \times 7 = 34596 \]
Now, divide both sides by \( 1408 \):
\[ H = \frac{34596}{1408} \]
Calculating \( \frac{34596}{1408} \):
\[ H = 24.5 \, \text{cm} \]

### Final Answer:
The height of water in the cylindrical vessel is \( 24.5 \, \text{cm} \).

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