Water flows at the rate of 10 m per minute from a cylindrical pipe 5 mm in diameter . A conical vessel whose diameter is 40 cm and depth 24 cm is filled. The time taken to fill the conical vessel is :

Options

A. 50 min
B. 50 min . 12 sec.
C. 51 min . 12 sec
D. 51 min . 12 sec

Answer

To solve the problem, we need to find the time taken to fill a conical vessel with water flowing from a cylindrical pipe. Here’s a step-by-step solution:

### 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 and \( h \) is the height of the cone.

Given:
- Diameter of the conical vessel = 40 cm, hence radius \( r = \frac{40}{2} = 20 \) cm
- Height \( h = 24 \) cm

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (20)^2 (24) \]
\[ V = \frac{1}{3} \pi (400)(24) \]
\[ V = \frac{1}{3} \pi (9600) \]
\[ V = 3200 \pi \text{ cm}^3 \]

### Step 2: Calculate the Flow Rate of Water from the Pipe
The flow rate of water from the cylindrical pipe is given as 10 m per minute. We need to convert this to cm per minute:
\[ 10 \text{ m/min} = 10 \times 100 = 1000 \text{ cm/min} \]

### Step 3: Calculate the Cross-sectional Area of the Pipe
The diameter of the pipe is given as 5 mm, which is equivalent to:
\[ 5 \text{ mm} = 0.5 \text{ cm} \]
Thus, the radius \( r_p \) of the pipe is:
\[ r_p = \frac{0.5}{2} = 0.25 \text{ cm} \]

The cross-sectional area \( A \) of the pipe is given by:
\[ A = \pi r_p^2 = \pi (0.25)^2 = \pi (0.0625) = 0.0625 \pi \text{ cm}^2 \]

### Step 4: Calculate the Volume of Water Flowing per Minute
The volume of water flowing out of the pipe per minute is:
\[ \text{Volume per minute} = \text{Area} \times \text{Flow Rate} \]
\[ = 0.0625 \pi \times 1000 \text{ cm}^3 \]
\[ = 62.5 \pi \text{ cm}^3 \]

### Step 5: Calculate the Time Taken to Fill the Conical Vessel
To find the time \( t \) taken to fill the conical vessel, we use the formula:
\[ t = \frac{\text{Volume of conical vessel}}{\text{Volume per minute}} \]
Substituting the values:
\[ t = \frac{3200 \pi}{62.5 \pi} \]
The \( \pi \) cancels out:
\[ t = \frac{3200}{62.5} = 51.2 \text{ minutes} \]

### Final Answer
The time taken to fill the conical vessel is **51.2 minutes**.

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