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 inflow and outflow of water in the vessel and establish the relationship between the maximum height of water (h) in the vessel, the rate of inflow (α), and the area of the hole (a).

### Step-by-Step Solution:

1. **Identify the Rate of Inflow**:
The water is being poured into the vessel at a constant rate, denoted as \( \alpha \) (in \( m^2/s \)). This means the volume of water entering the vessel per unit time is given by:
\[ \text{Rate of inflow} = \alpha \]

2. **Identify the Rate of Outflow**:
The water exits through a small hole at the bottom of the vessel. The rate of outflow can be expressed using Torricelli's law, which states that the speed of efflux (v) of a fluid under the force of gravity through an orifice is given by:
\[ v = \sqrt{2gh} \]
where \( g \) is the acceleration due to gravity and \( h \) is the height of the water column above the hole.

The rate of outflow (Q) can then be expressed as:
\[ \text{Rate of outflow} = a \cdot v = a \cdot \sqrt{2gh} \]

3. **Set Inflow Equal to Outflow**:
At equilibrium, the rate of inflow equals the rate of outflow:
\[ \alpha = a \cdot \sqrt{2gh} \]

4. **Solve for h**:
Rearranging the equation to solve for \( h \):
\[ \sqrt{2gh} = \frac{\alpha}{a} \]
Squaring both sides gives:
\[ 2gh = \left(\frac{\alpha}{a}\right)^2 \]
Now, isolating \( h \):
\[ h = \frac{\alpha^2}{2g a^2} \]

5. **Determine Proportional Relationships**:
From the equation \( h = \frac{\alpha^2}{2g a^2} \), we can see that:
- \( h \) is directly proportional to \( \alpha^2 \) (the rate of inflow).
- \( h \) is inversely proportional to \( a^2 \) (the area of the hole).

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

### Final Answer:
The maximum level of water \( h \) in the vessel is proportional to:
\[ h \propto \frac{\alpha^2}{a^2} \]