Water flows through a cylindrical pipe of diameter 5 mm at the rate of 10 m per minute and falls into a conical vessel having 40 cm as the diameter of its base and 24 cm as its height. How long will it take to fill the vessel ?

Options

A. `48` mins `15` secs
B. `51` mins `2` secs
C. `52` mins `1` sec
D. `51` mins `12` secs

Answer

To solve the problem step by step, we need to find out how long it will take to fill the conical vessel with water flowing from the cylindrical pipe.

### Step 1: Calculate the volume of the conical vessel.

The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone
- \( h \) is the height of the cone

Given:
- Diameter of the base of the cone = 40 cm, so the radius \( r = \frac{40}{2} = 20 \) cm
- Height of the cone \( h = 24 \) cm

Now substituting the values into the volume formula:

\[ V = \frac{1}{3} \pi (20)^2 (24) \]

Calculating \( (20)^2 = 400 \):

\[ V = \frac{1}{3} \pi (400) (24) \]

Calculating \( 400 \times 24 = 9600 \):

\[ V = \frac{1}{3} \pi (9600) \]

Calculating \( \frac{9600}{3} = 3200 \):

\[ V = 3200 \pi \text{ cm}^3 \]

### Step 2: Calculate the flow rate of water from the cylindrical pipe.

The formula for the volume of water flowing through a cylindrical pipe is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the pipe
- \( h \) is the height (or length) of water flowing through the pipe

Given:
- Diameter of the pipe = 5 mm, so the radius \( r = \frac{5}{2} = 2.5 \) mm = \( \frac{2.5}{10} = 0.25 \) cm
- The water flows at a rate of 10 m/min = 1000 cm/min

Now substituting the values into the volume formula:

\[ V = \pi (0.25)^2 (1000) \]

Calculating \( (0.25)^2 = 0.0625 \):

\[ V = \pi (0.0625) (1000) \]

Calculating \( 0.0625 \times 1000 = 62.5 \):

\[ V = 62.5 \pi \text{ cm}^3 \text{ per minute} \]

### Step 3: Calculate the time taken to fill the conical vessel.

To find the time \( t \) taken to fill the conical vessel, we can use the formula:

\[ t = \frac{\text{Volume of the cone}}{\text{Flow rate}} \]

Substituting the values we found:

\[ t = \frac{3200 \pi}{62.5 \pi} \]

The \( \pi \) cancels out:

\[ t = \frac{3200}{62.5} \]

Calculating \( \frac{3200}{62.5} = 51.2 \) minutes.

### Final Answer:
It will take **51.2 minutes** to fill the conical vessel.

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