A cylindrical tank has a holes of `3 cm^(2)` in its bottom. If the water is allowed to flow into the tank form a tube above it at the rate of `80 cm^(3)//sec`. Then the maximum height up to which water can rise in the tank is

Options

A. `1.1 cm`
B. `5 cm`
C. `10cm`
D. `0.25 cm`

Explanation

To solve the problem, we need to determine the maximum height of water that can rise in a cylindrical tank with a hole at the bottom, given the inflow and outflow rates.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Area of the hole at the bottom of the tank, \( A = 3 \, \text{cm}^2 \)
- Volume flow rate of water entering the tank, \( Q_{\text{in}} = 80 \, \text{cm}^3/\text{s} \)

2. **Establish the Relationship Between Inflow and Outflow**:
- At maximum height, the inflow rate must equal the outflow rate:
\[ Q_{\text{in}} = Q_{\text{out}} \]
- The outflow rate can be expressed as:
\[ Q_{\text{out}} = A \cdot v_{\text{efflux}} \]
- Where \( v_{\text{efflux}} \) is the velocity of water flowing out of the hole.

3. **Use Torricelli's Law to Find the Efflux Velocity**:
- According to Torricelli's law, the velocity of efflux is given by:
\[ v_{\text{efflux}} = \sqrt{2gh} \]
- Where \( h \) is the height of the water column above the hole and \( g \) is the acceleration due to gravity (approximately \( 980 \, \text{cm/s}^2 \)).

4. **Set Up the Equation**:
- Equating the inflow and outflow rates:
\[ Q_{\text{in}} = A \cdot \sqrt{2gh} \]
- Substituting the known values:
\[ 80 = 3 \cdot \sqrt{2gh} \]

5. **Solve for \( \sqrt{2gh} \)**:
- Rearranging gives:
\[ \sqrt{2gh} = \frac{80}{3} \]

6. **Square Both Sides**:
- Squaring both sides results in:
\[ 2gh = \left(\frac{80}{3}\right)^2 \]
- Calculating the right side:
\[ 2gh = \frac{6400}{9} \]

7. **Solve for Height \( h \)**:
- Rearranging for \( h \):
\[ h = \frac{6400}{18g} \]
- Substituting \( g = 980 \, \text{cm/s}^2 \):
\[ h = \frac{6400}{18 \cdot 980} \]
- Simplifying gives:
\[ h \approx \frac{6400}{17640} \approx 0.362 \, \text{cm} \]

### Final Answer:
The maximum height up to which water can rise in the tank is approximately \( 0.36 \, \text{cm} \).