A rectangular vessel when full of water takes 10 minutes to be emptied through an orifice in its bottom. How much time will it take to be emptied when half filled with water

Options

A. 3 min
B. 5 min
C. 7 min
D. 9 min

Explanation

To solve the problem of how long it will take to empty a rectangular vessel when it is half-filled with water, we can use the principle of fluid dynamics related to the time taken to empty a vessel through an orifice.

### Step-by-step Solution:

1. **Understand the relationship between height and time**:
The time \( T \) taken to empty a vessel of height \( h \) is given by the formula:
\[ T = k \sqrt{\frac{2h}{g}} \]
where \( k \) is a constant, \( g \) is the acceleration due to gravity, and \( h \) is the height of the water column.

2. **Given information**:
We know that when the vessel is full (height \( h \)), it takes 10 minutes to empty. Thus, we can write:
\[ T = 10 \text{ minutes} \]

3. **Determine the time for half-filled vessel**:
When the vessel is half-filled, the height of the water column is \( \frac{h}{2} \). We can denote the time taken to empty the half-filled vessel as \( T' \):
\[ T' = k \sqrt{\frac{2 \cdot \frac{h}{2}}{g}} = k \sqrt{\frac{h}{g}} \]

4. **Relate \( T' \) to \( T \)**:
From the original equation for \( T \):
\[ T = k \sqrt{\frac{2h}{g}} \]
We can express \( T' \) in terms of \( T \):
\[ T' = \frac{T}{\sqrt{2}} \]

5. **Substitute the known value of \( T \)**:
Since \( T = 10 \) minutes, we can substitute this into the equation:
\[ T' = \frac{10}{\sqrt{2}} \approx \frac{10}{1.414} \approx 7.07 \text{ minutes} \]

6. **Final answer**:
Therefore, the time taken to empty the vessel when it is half-filled is approximately:
\[ T' \approx 7 \text{ minutes} \]

### Summary:
The time taken to empty the vessel when half-filled with water is approximately 7 minutes.