Water flows at a rate of 10 m per minute through a cylindrical pipe having its diameter as 20 mm. How much time will it take to fill a conical vessel of base diameter 40 cm and depth 24 cm.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the diameter of the cylindrical pipe to radius in centimeters
- Given diameter of the cylindrical pipe = 20 mm
- Convert to centimeters: 20 mm = 2 cm
- Therefore, the radius \( r \) of the cylindrical pipe = \( \frac{2}{2} = 1 \) cm.

**Hint:** Remember that 1 cm = 10 mm, so to convert mm to cm, divide by 10.

### Step 2: Convert the rate of flow from meters per minute to centimeters per minute
- Given rate of flow = 10 m/min
- Convert to centimeters: \( 10 \, \text{m/min} = 10 \times 100 = 1000 \, \text{cm/min} \).

**Hint:** To convert meters to centimeters, multiply by 100.

### Step 3: Calculate the volume of water flowing through the cylindrical pipe in one minute
- The formula for the volume of a cylinder is \( V = \pi r^2 h \).
- Here, \( r = 1 \) cm and \( h = 1000 \) cm (the distance water flows in one minute).
- Volume of water in one minute = \( V = \pi (1)^2 (1000) = 1000\pi \, \text{cm}^3 \).

**Hint:** The height in this case is the distance the water travels in one minute.

### Step 4: Calculate the radius of the conical vessel
- Given diameter of the conical vessel = 40 cm
- Therefore, the radius \( R \) of the conical vessel = \( \frac{40}{2} = 20 \) cm.

**Hint:** The radius is half of the diameter.

### Step 5: Calculate the volume of the conical vessel
- The formula for the volume of a cone is \( V = \frac{1}{3} \pi R^2 h \).
- Here, \( R = 20 \) cm and \( h = 24 \) cm.
- Volume of the conical vessel = \( V = \frac{1}{3} \pi (20)^2 (24) = \frac{1}{3} \pi (400)(24) = \frac{9600}{3} \pi = 3200\pi \, \text{cm}^3 \).

**Hint:** Make sure to square the radius before multiplying by the height.

### Step 6: Calculate the time taken to fill the conical vessel
- Time taken = \( \frac{\text{Volume of conical vessel}}{\text{Volume of water through pipe in one minute}} \).
- Time taken = \( \frac{3200\pi}{1000\pi} = \frac{3200}{1000} = 3.2 \, \text{minutes} \).

**Hint:** The \( \pi \) cancels out, simplifying the calculation.

### Final Answer
The time taken to fill the conical vessel is **3.2 minutes**.