A single pipe of diameter x has to be replaced by six pipes of diameters 12 cm each. If the speed of flow of the liquid is maintained the same, then the value of x is

Options

A. 14.69 cm
B. 29.39 cm
C. 18.65 cm
D. 22.21 cm

Answer

To solve the problem, we need to find the diameter \( x \) of a single pipe that can be replaced by six pipes of diameter 12 cm each, while maintaining the same speed of flow of the liquid.

### Step-by-Step Solution:

1. **Calculate the Area of the Single Pipe:**
- The area \( A \) of a pipe with diameter \( x \) can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where \( r \) is the radius of the pipe. The radius \( r \) is half of the diameter:
\[ r = \frac{x}{2} \]
Therefore, the area of the single pipe is:
\[ A_1 = \pi \left(\frac{x}{2}\right)^2 = \pi \frac{x^2}{4} \]

2. **Calculate the Area of the Six Pipes:**
- Each of the six pipes has a diameter of 12 cm, so the radius of each pipe is:
\[ r = \frac{12}{2} = 6 \text{ cm} \]
The area of one of these pipes is:
\[ A_2 = \pi (6)^2 = \pi \cdot 36 \]
Since there are six pipes, the total area for the six pipes is:
\[ A_{total} = 6 \cdot A_2 = 6 \cdot \pi \cdot 36 = 216\pi \]

3. **Set the Areas Equal:**
- Since the speed of flow is maintained the same, the area of the single pipe must equal the total area of the six pipes:
\[ \pi \frac{x^2}{4} = 216\pi \]

4. **Cancel \( \pi \) from Both Sides:**
- Dividing both sides by \( \pi \):
\[ \frac{x^2}{4} = 216 \]

5. **Solve for \( x^2 \):**
- Multiply both sides by 4:
\[ x^2 = 216 \cdot 4 = 864 \]

6. **Calculate \( x \):**
- Take the square root of both sides to find \( x \):
\[ x = \sqrt{864} \]
- Simplifying \( \sqrt{864} \):
\[ x = \sqrt{144 \cdot 6} = 12\sqrt{6} \approx 29.39 \text{ cm} \]

### Final Answer:
The value of \( x \) is approximately **29.39 cm**.