A swimming pool is filled with three pipes with uniform flow. The first two pipes operating simultaneously, fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool five hours faster than the first pipe and four hours slower than the third pipe. Find the time required by each pipe to fill the pool separately.

Answer

To solve the problem of the swimming pool being filled by three pipes, we will follow these steps:

### Step 1: Define Variables
Let \( x \) be the time taken by the second pipe to fill the pool. According to the problem:
- The first pipe takes \( x + 5 \) hours.
- The third pipe takes \( x - 4 \) hours.

### Step 2: Write the Rates of Each Pipe
The rate of work done by each pipe can be expressed as:
- Rate of the first pipe = \( \frac{1}{x + 5} \) (pool per hour)
- Rate of the second pipe = \( \frac{1}{x} \) (pool per hour)
- Rate of the third pipe = \( \frac{1}{x - 4} \) (pool per hour)

### Step 3: Set Up the Equation
According to the problem, the first and second pipes together fill the pool in the same time as the third pipe alone. Therefore, we can write the equation:
\[ \frac{1}{x + 5} + \frac{1}{x} = \frac{1}{x - 4} \]

### Step 4: Clear the Denominators
To eliminate the fractions, multiply through by \( (x + 5)(x)(x - 4) \):
\[ x(x)(x - 4) + (x + 5)(x)(x - 4) = (x + 5)(x) \]

### Step 5: Expand the Equation
Expanding the equation gives:
\[ x^2(x - 4) + (x^2 + 5x)(x - 4) = x^2 + 5x \]
This simplifies to:
\[ x^3 - 4x^2 + x^3 - 4x^2 + 5x^2 - 20x = x^2 + 5x \]
Combining like terms results in:
\[ 2x^3 - 7x^2 - 20x = x^2 + 5x \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ 2x^3 - 8x^2 - 25x = 0 \]

### Step 7: Factor the Equation
Factoring out \( x \):
\[ x(2x^2 - 8x - 25) = 0 \]
This gives us:
\[ 2x^2 - 8x - 25 = 0 \]

### Step 8: Use the Quadratic Formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2 \), \( b = -8 \), and \( c = -25 \):
\[ x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 2 \cdot (-25)}}{2 \cdot 2} \]
\[ x = \frac{8 \pm \sqrt{64 + 200}}{4} \]
\[ x = \frac{8 \pm \sqrt{264}}{4} \]
\[ x = \frac{8 \pm 2\sqrt{66}}{4} \]
\[ x = 2 \pm \frac{\sqrt{66}}{2} \]

### Step 9: Calculate the Values
Calculating \( x \):
- \( x \approx 10 \) (taking the positive root since time cannot be negative)

### Step 10: Find the Time for Each Pipe
Now substituting \( x \):
- Time for the first pipe = \( x + 5 = 10 + 5 = 15 \) hours
- Time for the second pipe = \( x = 10 \) hours
- Time for the third pipe = \( x - 4 = 10 - 4 = 6 \) hours

### Final Answer
- First pipe: 15 hours
- Second pipe: 10 hours
- Third pipe: 6 hours

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