Which of these pipes will empty a pool the fastest? i. One pipe of diameter 60 cm ii. Two pipes of diameter 30 cm each iii. Three pipes of diameter 20 cm each

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To determine which pipe will empty a pool the fastest, we need to calculate the cross-sectional area of each pipe since the flow rate of water through a pipe is proportional to its cross-sectional area.

### Step-by-Step Solution:

1. **Calculate the area of the first pipe (60 cm diameter)**:
- The radius \( r \) is half of the diameter:
\[ r = \frac{60 \text{ cm}}{2} = 30 \text{ cm} \]
- The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
- Substituting the radius:
\[ A_1 = \pi (30 \text{ cm})^2 = \pi (900 \text{ cm}^2) = 900\pi \text{ cm}^2 \]

2. **Calculate the area of the second option (two pipes of 30 cm diameter each)**:
- The radius for each pipe is:
\[ r = \frac{30 \text{ cm}}{2} = 15 \text{ cm} \]
- The area of one pipe is:
\[ A = \pi (15 \text{ cm})^2 = \pi (225 \text{ cm}^2) = 225\pi \text{ cm}^2 \]
- Since there are two pipes, the total area is:
\[ A_2 = 2 \times 225\pi \text{ cm}^2 = 450\pi \text{ cm}^2 \]

3. **Calculate the area of the third option (three pipes of 20 cm diameter each)**:
- The radius for each pipe is:
\[ r = \frac{20 \text{ cm}}{2} = 10 \text{ cm} \]
- The area of one pipe is:
\[ A = \pi (10 \text{ cm})^2 = \pi (100 \text{ cm}^2) = 100\pi \text{ cm}^2 \]
- Since there are three pipes, the total area is:
\[ A_3 = 3 \times 100\pi \text{ cm}^2 = 300\pi \text{ cm}^2 \]

4. **Compare the areas**:
- Area of the first pipe: \( 900\pi \text{ cm}^2 \)
- Area of the second option: \( 450\pi \text{ cm}^2 \)
- Area of the third option: \( 300\pi \text{ cm}^2 \)

5. **Conclusion**:
- The first pipe has the largest cross-sectional area, therefore it will empty the pool the fastest.

### Final Answer:
The first pipe (diameter 60 cm) will empty the pool the fastest.