A water tank has a hole in its wall at a distance of 40 m below the free surface of water. Compute the velocity of flow of water from the hole. If the radius of the hole is 1 mm. find the rate of flow of water

Options

A. `26m//s,8.8xx10^(-5)m^(3)//s`
B. `28m//s,8.8xx10^(-5)m^(3)//s`
C. `28m//s,6.8xx10^(-5)m^(3)//5`
D. `26m//s, 9.8xx10^(-5)m^(3)//s`

Answer

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

### Step 1: Identify the parameters
We are given:
- Depth of the hole (h) = 40 m
- Radius of the hole (r) = 1 mm = 1 × 10^-3 m
- Acceleration due to gravity (g) = 9.8 m/s²

### Step 2: Apply Torricelli's theorem
According to Torricelli's theorem, the velocity (V) of fluid flowing out of a hole under the influence of gravity is given by the formula:
\[ V = \sqrt{2gh} \]

### Step 3: Substitute the values into the equation
Now, substituting the values into the equation:
\[ V = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 40 \, \text{m}} \]

### Step 4: Calculate the value inside the square root
Calculating the value:
\[ 2 \times 9.8 \times 40 = 784 \]

### Step 5: Take the square root
Now, taking the square root:
\[ V = \sqrt{784} = 28 \, \text{m/s} \]

### Step 6: Calculate the cross-sectional area of the hole
The cross-sectional area (A) of the hole can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Substituting the radius:
\[ A = \pi (1 \times 10^{-3})^2 = \pi \times 10^{-6} \, \text{m}^2 \]

### Step 7: Calculate the flow rate (Q)
The flow rate (Q) can be calculated using the formula:
\[ Q = A \times V \]
Substituting the values:
\[ Q = \pi \times 10^{-6} \times 28 \]

### Step 8: Calculate the flow rate
Calculating the flow rate:
\[ Q \approx 3.14 \times 10^{-6} \times 28 \approx 87.92 \times 10^{-6} \, \text{m}^3/\text{s} \]

### Step 9: Final result
Rounding the result gives:
\[ Q \approx 8.8 \times 10^{-5} \, \text{m}^3/\text{s} \]

### Summary of Results
- Velocity of water flowing from the hole: **28 m/s**
- Rate of flow of water: **8.8 × 10^-5 m³/s**