A cylindrical vessel of 92 cm height is kept filled upto to brim. It has four holes 1,2,3 and 4 which are respectively at heights of 20cm, 30cm, 46cm and 80cm from the horizontal floor. The water falling at the maximum horizontal distance from the vessel comes from :

Options

A. hole 4
B. hole 3
C. hole 2
D. hole 1

Answer

To solve the problem, we need to determine from which hole the water will fall at the maximum horizontal distance when it exits the cylindrical vessel. The key to this problem lies in understanding the relationship between the height of the holes and the range of the water jet.

### Step-by-Step Solution:

1. **Identify the Heights of the Holes**:
- The heights of the holes from the floor are:
- Hole 1: 20 cm
- Hole 2: 30 cm
- Hole 3: 46 cm
- Hole 4: 80 cm

2. **Understand the Efflux Velocity**:
- The efflux velocity of water coming out of a hole at a height \( h \) from the surface of the water is given by the formula:
\[ v = \sqrt{2g(92 - h)} \]
- Here, \( g \) is the acceleration due to gravity, and \( 92 \) cm is the total height of the vessel.

3. **Determine the Time of Flight**:
- The time of flight \( t \) for the water to fall from height \( h \) to the ground can be calculated using the kinematic equation:
\[ h = \frac{1}{2} g t^2 \]
- Rearranging gives:
\[ t = \sqrt{\frac{2h}{g}} \]

4. **Calculate the Horizontal Range**:
- The horizontal range \( R \) can be calculated using the formula:
\[ R = v \cdot t \]
- Substituting the expressions for \( v \) and \( t \):
\[ R = \sqrt{2g(92 - h)} \cdot \sqrt{\frac{2h}{g}} = 2\sqrt{h(92 - h)} \]

5. **Maximize the Range**:
- To find the height \( h \) that maximizes the range \( R \), we need to maximize the expression \( R^2 = 4h(92 - h) \).
- This is a quadratic function in terms of \( h \), which opens downwards (since the coefficient of \( h^2 \) is negative).

6. **Find the Maximum Value**:
- The maximum value occurs at:
\[ h = \frac{92}{2} = 46 \text{ cm} \]
- This corresponds to Hole 3.

7. **Conclusion**:
- The water falling at the maximum horizontal distance from the vessel comes from Hole 3, which is at a height of 46 cm.

### Final Answer:
The water falling at the maximum horizontal distance comes from **Hole 3** (46 cm).