Which of these pipes will empty a pool the fastest?
1. One pipe of diameter 60 cm 2. Two pipes of diameter 30 cm each 3. 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 the pool the fastest, we need to calculate the cross-sectional area of each pipe. The larger the area, the faster the water can flow through the pipe.

### Step-by-Step Solution:

**Step 1: Calculate the area of the first pipe (diameter = 60 cm)**
- The radius \( r \) of the pipe 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 into the formula:
\[ A = \pi (30 \text{ cm})^2 = \pi (900 \text{ cm}^2) = 900\pi \text{ cm}^2 \]

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

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

**Step 4: Compare the areas**
- First pipe: \( 900\pi \text{ cm}^2 \)
- Two pipes: \( 450\pi \text{ cm}^2 \)
- Three pipes: \( 300\pi \text{ cm}^2 \)

From the calculations, we see that the first pipe has the largest area.

### Conclusion:
The pipe with a diameter of 60 cm will empty the pool the fastest.

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