If two pipes function simultaneously, a tank is filled in 12 hours. One pipe fills the tank 10 hours faster than the other. How many hours does the faster pipe alone take to fill the tank?

Options

A. 20 hrs
B. 18 hrs
C. 15 hrs
D. 12 hrs

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define Variables
Let the time taken by the slower pipe to fill the tank be \( x \) hours. Then, the faster pipe will take \( x - 10 \) hours to fill the tank, as it fills the tank 10 hours faster than the slower pipe.

**Hint:** Start by defining variables for the unknowns in the problem.

### Step 2: Write the Rate of Work
The rate of work for the slower pipe is \( \frac{1}{x} \) (tank per hour), and the rate of work for the faster pipe is \( \frac{1}{x - 10} \) (tank per hour).

**Hint:** Remember that the rate of work is the reciprocal of the time taken to fill the tank.

### Step 3: Combine the Rates
When both pipes are working together, their combined rate is:
\[ \frac{1}{x} + \frac{1}{x - 10} \]
This combined rate fills the tank in 12 hours, so we can set up the equation:
\[ \frac{1}{x} + \frac{1}{x - 10} = \frac{1}{12} \]

**Hint:** Use the concept of combined rates to form an equation.

### Step 4: Clear the Denominators
To eliminate the fractions, multiply through by \( 12x(x - 10) \):
\[ 12(x - 10) + 12x = x(x - 10) \]

**Hint:** Multiplying by the least common multiple helps to simplify the equation.

### Step 5: Simplify the Equation
Expanding both sides gives:
\[ 12x - 120 + 12x = x^2 - 10x \]
Combine like terms:
\[ 24x - 120 = x^2 - 10x \]
Rearranging gives:
\[ x^2 - 34x + 120 = 0 \]

**Hint:** Rearranging the equation into standard quadratic form is crucial for solving it.

### Step 6: Factor the Quadratic
Now, we need to factor the quadratic equation:
\[ x^2 - 34x + 120 = 0 \]
This factors to:
\[ (x - 30)(x - 4) = 0 \]

**Hint:** Look for two numbers that multiply to the constant term and add to the coefficient of \( x \).

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
\[ x - 30 = 0 \quad \Rightarrow \quad x = 30 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
Since \( x \) represents the time taken by the slower pipe, we discard \( x = 4 \) because the faster pipe would then take negative time.

**Hint:** Always check the context of the solution to ensure it makes sense.

### Step 8: Find the Time for the Faster Pipe
The time taken by the faster pipe is:
\[ x - 10 = 30 - 10 = 20 \text{ hours} \]

**Hint:** The final step is to substitute back to find the value of the faster pipe.

### Final Answer
The faster pipe alone takes **20 hours** to fill the tank.