Water is being poured in a vessel at a constant rate `alpha m^(2)//s`. There is a small hole of area a at the bottom of the tank. The maximum level of water in the vessel is proportional to

Options

A. `alpha`
B. `alpha^(2)`
C. `a^(-1)`
D. `a^(-2)`

Answer

To solve the problem, we need to analyze the situation where water is being poured into a vessel at a constant rate while simultaneously leaking out through a small hole at the bottom.

### Step-by-Step Solution:

1. **Understand the Flow Rate**:
- Water is being poured into the vessel at a constant rate, denoted as \( \alpha \) m³/s.
- This means the volume of water entering the vessel per second is constant.

2. **Identify the Hole's Effect**:
- There is a small hole at the bottom of the tank with an area \( a \).
- Water will flow out of this hole due to the pressure difference between the water inside the tank and the atmospheric pressure outside.

3. **Apply Torricelli's Law**:
- According to Torricelli's Law, the speed \( v \) of efflux of a fluid under the force of gravity through an orifice is given by:
\[ v = \sqrt{2gh} \]
- Here, \( h \) is the height of the water column above the hole, and \( g \) is the acceleration due to gravity.

4. **Calculate the Outflow Rate**:
- The volume flow rate \( Q \) through the hole can be expressed as:
\[ Q = a \cdot v = a \cdot \sqrt{2gh} \]
- This represents the volume of water leaving the vessel per second through the hole.

5. **Establish the Equilibrium Condition**:
- At equilibrium, the rate at which water is being poured into the vessel equals the rate at which water is flowing out:
\[ \alpha = a \cdot \sqrt{2gh} \]

6. **Rearranging the Equation**:
- To find the maximum height \( h \) of the water in the vessel, we can rearrange the equation:
\[ \sqrt{2gh} = \frac{\alpha}{a} \]
- Squaring both sides gives:
\[ 2gh = \left(\frac{\alpha}{a}\right)^2 \]

7. **Solving for Height \( h \)**:
- Now, isolate \( h \):
\[ h = \frac{\alpha^2}{2g a^2} \]

8. **Determine Proportionality**:
- From the equation \( h = \frac{\alpha^2}{2g a^2} \), we can see that \( h \) is proportional to \( \alpha^2 \) when \( g \) and \( a \) are constants.

### Conclusion:
The maximum level of water in the vessel is proportional to \( \alpha^2 \).