Let's denote the time taken by the first pipe to fill the tank as $x$ hours. According to the problem, the second pipe takes 5 hours less than the first pipe, so it takes $x - 5$ hours. The third pipe takes 4 hours less than the second pipe, so it takes $(x - 5) - 4 = x - 9$ hours.

Step-by-Step Solution:

1. Work Done by Each Pipe:

   • First pipe: $\text{Work rate} = \frac{1}{x}$
   • Second pipe: $\text{Work rate} = \frac{1}{x - 5}$
   • Third pipe: $\text{Work rate} = \frac{1}{x - 9}$

2. Combined Work Rate of the First and Second Pipes: Since the first two pipes operating together take the same time to fill the tank as the third pipe alone: $\frac{1}{x} + \frac{1}{x - 5} = \frac{1}{x - 9}$

3. Solve the Equation: To solve this equation, we'll first find a common denominator on the left-hand side: $\frac{(x - 5) + x}{x(x - 5)} = \frac{1}{x - 9}$ Simplifying: $\frac{2x - 5}{x(x - 5)} = \frac{1}{x - 9}$

   Cross-multiplying to eliminate the denominators: $(2x - 5)(x - 9) = x(x - 5)$

   Expand both sides: $2x^2 - 18x - 5x + 45 = x^2 - 5x$ Simplify: $2x^2 - 23x + 45 = x^2 - 5x$ Move all terms to one side: $x^2 - 18x + 45 = 0$

4. Solve the Quadratic Equation: We can solve the quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = -18$, and $c = 45$. $x = \frac{18 \pm \sqrt{(-18)^2 - 4(1)(45)}}{2(1)}$ Simplify: $x = \frac{18 \pm \sqrt{324 - 180}}{2} = \frac{18 \pm \sqrt{144}}{2} = \frac{18 \pm 12}{2}$ This gives two possible solutions: $x = \frac{18 + 12}{2} = \frac{30}{2} = 15 \quad \text{or} \quad x = \frac{18 - 12}{2} = \frac{6}{2} = 3$

5. Check the Validity: Since the second pipe takes $x - 5$ hours and $x = 15$ gives a positive time, $x = 15$ is the correct answer (because $x = 3$ would give a negative or unrealistic time for the second pipe).

The correct answer is C. 15 hours.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you determine the combined work rate of two or more pipes working together?
2. What is the quadratic formula, and how is it used to solve equations?
3. How can this type of problem be solved using algebraic substitution?
4. Can you solve similar problems involving three or more variables?
5. What are some practical applications of work-rate problems in real life?

Tip: For work-rate problems, always express the work rates of each individual component and combine them accordingly. This makes setting up and solving equations easier.