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. `10 m^(3) s^(-1)`
D. `10^(-4) m^(3) s^(-1)`

Answer

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

### Step 1: Understand the problem
We have a tank filled with water to a height of 5 meters, and there is a hole at the bottom with an area of 10 cm². We need to find the rate at which water leaks out of this hole.

### Step 2: Convert the area of the hole to square meters
The area of the hole is given in cm², so we need to convert it to m² for consistency with the SI unit system.
\[ \text{Area} = 10 \, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 10^{-4} \, \text{m}^2 \]

### Step 3: Use Torricelli’s theorem to find the velocity of efflux
According to Torricelli’s theorem, the velocity \( v \) of efflux of a fluid under the force of gravity through a hole is given by:
\[ v = \sqrt{2gh} \]
where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)) and \( h \) is the height of the fluid above the hole. In this case, \( h = 5 \, \text{m} \).

Calculating the velocity:
\[ v = \sqrt{2 \times 9.81 \, \text{m/s}^2 \times 5 \, \text{m}} = \sqrt{98.1} \approx 9.9 \, \text{m/s} \]

### Step 4: Calculate the rate of leakage (Q)
The rate of leakage \( Q \) can be calculated using the formula:
\[ Q = \text{Area} \times \text{Velocity} \]
Substituting the values we have:
\[ Q = (10^{-4} \, \text{m}^2) \times (9.9 \, \text{m/s}) = 9.9 \times 10^{-4} \, \text{m}^3/\text{s} \]

### Step 5: Express the result
Thus, the rate of leakage of water from the hole is:
\[ Q \approx 9.9 \times 10^{-4} \, \text{m}^3/\text{s} \]

### Final Answer
The rate of leakage of water from the hole is approximately \( 9.9 \times 10^{-4} \, \text{m}^3/\text{s} \).

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