Water flows at the rate of 10 m/min from a cylindrical pipe 5 mm in diameter. How long will it take to fill up a conical vessel whose diameter at the base is 40 cm and depth is 24 cm?

Options

A. 51 min 12 s
B. 52 min 1 s
C. 48 min 15 s
D. 55 min

Answer

To solve the problem, we need to follow these steps:

### Step 1: Find 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, thus radius \( r = \frac{40}{2} = 20 \) cm.
- Height \( h = 24 \) cm.

Substituting the values into the volume formula:

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

Calculating \( V \):

\[ V = \frac{1}{3} \pi (400) (24) = \frac{1}{3} \pi (9600) = 3200\pi \text{ cm}^3 \]

### Step 2: Convert the volume to cubic meters.
Since the flow rate is given in meters, we need to convert the volume from cubic centimeters to cubic meters:

\[ 1 \text{ m}^3 = 1,000,000 \text{ cm}^3 \]

Thus,

\[ 3200\pi \text{ cm}^3 = \frac{3200\pi}{1000000} \text{ m}^3 \approx 0.01005309649 \text{ m}^3 \]

### Step 3: Find the flow rate of water from the pipe.
The diameter of the pipe is 5 mm, so the radius \( r \) is:

\[ r = \frac{5}{2} = 2.5 \text{ mm} = 0.0025 \text{ m} \]

The cross-sectional area \( A \) of the pipe is:

\[ A = \pi r^2 = \pi (0.0025)^2 = \pi (0.00000625) \approx 0.00001963495408 \text{ m}^2 \]

The flow rate \( Q \) is given by:

\[ Q = A \times \text{velocity} \]

Given the velocity of water flow is 10 m/min, we need to convert this to meters per second:

\[ 10 \text{ m/min} = \frac{10}{60} \text{ m/s} \approx 0.1667 \text{ m/s} \]

Now, substituting the values into the flow rate formula:

\[ Q = 0.00001963495408 \text{ m}^2 \times 0.1667 \text{ m/s} \approx 0.000003272 \text{ m}^3/s \]

### Step 4: Calculate the time taken to fill the conical vessel.
The time \( t \) taken to fill the volume \( V \) is given by:

\[ t = \frac{V}{Q} \]

Substituting the values:

\[ t = \frac{0.01005309649 \text{ m}^3}{0.000003272 \text{ m}^3/s} \approx 3074.57 \text{ seconds} \]

To convert seconds into minutes:

\[ t \approx \frac{3074.57}{60} \approx 51.24 \text{ minutes} \]

### Final Answer:
It will take approximately **51.24 minutes** to fill up the conical vessel.

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