Water running in a cylindrical pipe of inner diameter 7 cm, is collected in a container at the rate of 192.5 litres per minute. Find the rate flow of water in the pipe in km/hr.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the flow rate from liters per minute to cubic centimeters per hour.
Given that the flow rate is 192.5 liters per minute, we know that:
1 liter = 1000 cubic centimeters (cm³).
Therefore, we convert liters to cubic centimeters:
\[ 192.5 \text{ liters/minute} = 192.5 \times 1000 \text{ cm}^3/\text{minute} \]
Now, converting this to cubic centimeters per hour:
\[ 192.5 \times 1000 \text{ cm}^3/\text{minute} \times 60 \text{ minutes/hour} = 11550000 \text{ cm}^3/\text{hour} \]

### Step 2: Calculate the radius of the cylindrical pipe.
The inner diameter of the pipe is given as 7 cm. Therefore, the radius (r) is:
\[ r = \frac{\text{diameter}}{2} = \frac{7 \text{ cm}}{2} = 3.5 \text{ cm} \]

### Step 3: Calculate the cross-sectional area of the cylindrical pipe.
The cross-sectional area (A) of the cylinder can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the value of r:
\[ A = \pi \left(3.5\right)^2 = \pi \times 12.25 \text{ cm}^2 \]
Using \(\pi \approx \frac{22}{7}\):
\[ A \approx \frac{22}{7} \times 12.25 \approx 38.5 \text{ cm}^2 \]

### Step 4: Find the rate of flow of water in the pipe.
Using the formula for flow rate:
\[ \text{Flow rate} = \frac{\text{Volume}}{\text{Time}} = A \times v \]
Where \(v\) is the velocity of water in the pipe. We can rearrange this to find \(v\):
\[ v = \frac{\text{Volume}}{A} \]
Substituting the values we have:
\[ v = \frac{11550000 \text{ cm}^3/\text{hour}}{38.5 \text{ cm}^2} \]
Calculating this gives:
\[ v \approx 300000 \text{ cm/hour} \]

### Step 5: Convert the velocity from cm/hour to km/hour.
To convert from centimeters per hour to kilometers per hour:
\[ \text{Velocity in km/hour} = \frac{v \text{ in cm/hour}}{100000} \]
Thus,
\[ \text{Velocity} = \frac{300000 \text{ cm/hour}}{100000} = 3 \text{ km/hour} \]

### Final Answer:
The rate of flow of water in the pipe is **3 km/hour**.

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