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. `9` minute
B. `7` minute
C. `5` minute
D. `3` minute

Answer

To solve the problem of how long it will take to empty a rectangular vessel when it is half filled with water, we can follow these steps:

### Step 1: Understand the relationship between height and time
The time taken to empty a vessel through an orifice is related to the height of the water in the vessel. The formula for the time \( T \) to empty the vessel is given by:

\[ T \propto \sqrt{h} \]

where \( h \) is the height of the water.

### Step 2: Define the initial conditions
Given that when the vessel is full (height \( h \)), it takes \( T_1 = 10 \) minutes to empty.

### Step 3: Set up the relationship for half-filled condition
When the vessel is half-filled, the height of the water is \( h/2 \). We need to find the time \( T_2 \) it takes to empty the vessel in this condition.

### Step 4: Use the proportionality to find the new time
Using the proportionality we established:

\[ \frac{T_1}{T_2} = \sqrt{\frac{h}{h/2}} \]

This simplifies to:

\[ \frac{T_1}{T_2} = \sqrt{2} \]

### Step 5: Substitute the known value of \( T_1 \)
Now substituting \( T_1 = 10 \) minutes into the equation:

\[ \frac{10}{T_2} = \sqrt{2} \]

### Step 6: Solve for \( T_2 \)
Rearranging the equation gives:

\[ T_2 = \frac{10}{\sqrt{2}} = 10 \times \frac{1}{\sqrt{2}} = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \]

### Step 7: Calculate the numerical value
Calculating \( 5\sqrt{2} \):

\[ 5\sqrt{2} \approx 5 \times 1.414 \approx 7.07 \text{ minutes} \]

### Final Answer
Thus, the time taken to empty the vessel when it is half filled with water is approximately **7.07 minutes**.

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