The level of water in a tank is 5 m high . A hole of the area `10 cm^2` is made in the bottom of the tank . The rate of leakage of water from the hole is

Options

A. `10^(-2)m^(3)s^(-1)`
B. `10^(2)m^(3)s^(-1)`
C. `10m^(3)s^(-1)`
D. `10^(-4)m^(-3)s^(-1)`

Answer

To solve the problem of finding the rate of leakage of water from a hole in a tank, we can follow these steps:

### Step 1: Understand the given information
We have a tank filled with water to a height of \( h = 5 \, \text{m} \). There is a hole at the bottom of the tank with an area \( A = 10 \, \text{cm}^2 \).

### Step 2: Convert the area into appropriate units
Since we need to work with SI units, we convert the area from \( \text{cm}^2 \) to \( \text{m}^2 \):
\[ A = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 10^{-3} \, \text{m}^2 \]

### Step 3: Use Torricelli's Law to find the velocity of water
According to Torricelli's Law, the velocity \( v \) of the water flowing out of the hole can be calculated using the formula:
\[ v = \sqrt{2gh} \]
where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)) and \( h \) is the height of the water column.

Substituting the values:
\[ v = \sqrt{2 \times 10 \, \text{m/s}^2 \times 5 \, \text{m}} = \sqrt{100} = 10 \, \text{m/s} \]

### Step 4: Calculate the flow rate (Q)
The flow rate \( Q \) (the volume of water leaking per second) can be calculated using the formula:
\[ Q = A \times v \]
Substituting the values we have:
\[ Q = 10^{-3} \, \text{m}^2 \times 10 \, \text{m/s} = 10^{-2} \, \text{m}^3/\text{s} \]

### Step 5: Conclusion
The rate of leakage of water from the hole is:
\[ Q = 10^{-2} \, \text{m}^3/\text{s} \]

### Final Answer
The rate of leakage of water from the hole is \( 10^{-2} \, \text{m}^3/\text{s} \).

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