Water flows at the rate of 5 m per min from a cylindrical pipe 16 mm in diameter. How long will it take to fill up a conical vessel whose radius is 12 cm and depth is 35 cm?

Options

A. 315 s
B. 365 s
C. 5 min
D. none of these

Answer

To solve the problem, we will follow these steps:

### 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 cone
- \( h \) is the height of the cone

Given:
- Radius \( r = 12 \) cm
- Height \( h = 35 \) cm

Substituting the values into the formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times (12)^2 \times 35 \]

Calculating \( (12)^2 \):

\[ (12)^2 = 144 \]

Now substituting back:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 144 \times 35 \]

Calculating \( 144 \times 35 \):

\[ 144 \times 35 = 5040 \]

Now substituting this value back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 5040 \]

Calculating \( \frac{5040}{3} \):

\[ \frac{5040}{3} = 1680 \]

Now substituting this value:

\[ V = \frac{22}{7} \times 1680 \]

Calculating \( \frac{22 \times 1680}{7} \):

\[ 22 \times 1680 = 36960 \]
\[ \frac{36960}{7} = 5280 \text{ cm}^3 \]

So, the volume of the conical vessel is \( 5280 \) cm³.

### Step 2: Calculate the Flow Rate of Water from the Pipe

The diameter of the cylindrical pipe is given as \( 16 \) mm. First, we convert this to centimeters:

\[ 16 \text{ mm} = 1.6 \text{ cm} \]

Now, the radius \( r \) of the pipe is:

\[ r = \frac{1.6}{2} = 0.8 \text{ cm} \]

The area \( A \) of the circular cross-section of the pipe is given by:

\[ A = \pi r^2 \]

Substituting the radius:

\[ A = \frac{22}{7} \times (0.8)^2 \]

Calculating \( (0.8)^2 \):

\[ (0.8)^2 = 0.64 \]

Now substituting back:

\[ A = \frac{22}{7} \times 0.64 \]

Calculating \( 22 \times 0.64 \):

\[ 22 \times 0.64 = 14.08 \]
\[ A = \frac{14.08}{7} = 2.01 \text{ cm}^2 \]

The flow rate of water is given as \( 5 \) m/min, which we convert to cm/min:

\[ 5 \text{ m/min} = 500 \text{ cm/min} \]

Now, the volume of water flowing per minute is:

\[ \text{Volume flow rate} = A \times \text{velocity} \]
\[ = 2.01 \text{ cm}^2 \times 500 \text{ cm/min} = 1005 \text{ cm}^3/\text{min} \]

### Step 3: Calculate the Time to Fill the Conical Vessel

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

\[ t = \frac{\text{Volume of the conical vessel}}{\text{Volume flow rate}} \]

Substituting the values:

\[ t = \frac{5280 \text{ cm}^3}{1005 \text{ cm}^3/\text{min}} \]

Calculating:

\[ t \approx 5.25 \text{ minutes} \]

### Final Answer

It will take approximately **5.25 minutes** to fill the conical vessel.

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