Water is flowing at the rate of 3km/hr through a circular pipe of 20 cm internal diameter into a circular cistern of diameter 10 m and depth 2m. In how much time will the cistern be filled ?

Options

A. 1 hour
B. 1 hour 40 minutes
C. 1 hour 20 minutes
D. 2 hours 40 minutes

Answer

To solve the problem, we need to find out how long it will take to fill a circular cistern using water flowing through a circular pipe. Let's break down the steps:

### Step 1: Find the volume of the cistern.
The formula for the volume \( V \) of a cylinder (which is the shape of the cistern) is given by:
\[ V = \pi r^2 h \]
Where:
- \( r \) = radius of the base of the cylinder
- \( h \) = height of the cylinder

The diameter of the cistern is 10 m, so the radius \( r \) is:
\[ r = \frac{10}{2} = 5 \text{ m} \]
The depth (height) \( h \) of the cistern is 2 m. Now, we can calculate the volume:
\[ V = \pi (5)^2 (2) = \pi (25)(2) = 50\pi \text{ cubic meters} \]

### Step 2: Find the volume of water flowing through the pipe in one hour.
The diameter of the pipe is 20 cm, so the radius \( r \) is:
\[ r = \frac{20}{2} = 10 \text{ cm} = 0.1 \text{ m} \]
The flow rate of water is 3 km/hr, which we convert to meters per second:
\[ 3 \text{ km/hr} = \frac{3000 \text{ m}}{3600 \text{ s}} = \frac{5}{6} \text{ m/s} \]

Now, we can find the volume of water flowing through the pipe in one second:
\[ \text{Area of cross-section of pipe} = \pi r^2 = \pi (0.1)^2 = 0.01\pi \text{ square meters} \]
Thus, the volume of water flowing through the pipe in one second is:
\[ \text{Volume per second} = \text{Area} \times \text{Flow rate} = 0.01\pi \times \frac{5}{6} = \frac{5\pi}{600} \text{ cubic meters} \]

### Step 3: Calculate the total volume of water flowing in one hour.
In one hour (3600 seconds), the total volume of water flowing through the pipe is:
\[ \text{Total volume in one hour} = \frac{5\pi}{600} \times 3600 = 30\pi \text{ cubic meters} \]

### Step 4: Calculate the time taken to fill the cistern.
Now, we need to find out how long it will take to fill the cistern with a volume of \( 50\pi \) cubic meters using a flow of \( 30\pi \) cubic meters per hour:
\[ \text{Time} = \frac{\text{Volume of cistern}}{\text{Volume flow rate}} = \frac{50\pi}{30\pi} = \frac{50}{30} = \frac{5}{3} \text{ hours} \]

### Step 5: Convert hours into hours and minutes.
\(\frac{5}{3}\) hours is equal to 1 hour and \(\frac{2}{3}\) of an hour. To convert \(\frac{2}{3}\) of an hour into minutes:
\[ \frac{2}{3} \times 60 = 40 \text{ minutes} \]
Thus, the total time to fill the cistern is:
\[ 1 \text{ hour } 40 \text{ minutes} \]

### Final Answer:
The cistern will be filled in **1 hour and 40 minutes**.

---