Pipes A and B can fill a tank in 18 minutes and 27 minutes ,respectively .C is an outlet pipe .When A,B and C are opened together ,the empty tank is completely filled in 54 minutes . Pipe C alone can empty the full tank in :

Options

A. `14(1)/(2)` minutes
B. 14 minutes
C. `13(1)/(2)` minutes
D. 13 minutes

Answer

To solve the problem step by step, we will first determine the rates at which pipes A, B, and C work, and then find out how long it takes for pipe C to empty the tank.

### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 18 minutes. Therefore, its rate of work is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{18 \text{ minutes}} = \frac{1}{18} \text{ tanks per minute} \]

- Pipe B can fill the tank in 27 minutes. Therefore, its rate of work is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{27 \text{ minutes}} = \frac{1}{27} \text{ tanks per minute} \]

### Step 2: Calculate the combined rate of A and B
To find the combined rate of A and B, we add their rates:
\[ \text{Rate of A + Rate of B} = \frac{1}{18} + \frac{1}{27} \]

To add these fractions, we need a common denominator. The least common multiple of 18 and 27 is 54.
\[ \frac{1}{18} = \frac{3}{54}, \quad \frac{1}{27} = \frac{2}{54} \]
Thus,
\[ \text{Rate of A + Rate of B} = \frac{3}{54} + \frac{2}{54} = \frac{5}{54} \text{ tanks per minute} \]

### Step 3: Determine the combined rate of A, B, and C
When pipes A, B, and C are opened together, they fill the tank in 54 minutes. Therefore, their combined rate is:
\[ \text{Rate of A + B + C} = \frac{1 \text{ tank}}{54 \text{ minutes}} = \frac{1}{54} \text{ tanks per minute} \]

### Step 4: Set up the equation for pipe C
From the previous steps, we have:
\[ \text{Rate of A + B + C} = \text{Rate of A + Rate of B} - \text{Rate of C} \]
Substituting the values we have:
\[ \frac{1}{54} = \frac{5}{54} - \text{Rate of C} \]

### Step 5: Solve for the rate of pipe C
Rearranging the equation gives:
\[ \text{Rate of C} = \frac{5}{54} - \frac{1}{54} = \frac{4}{54} = \frac{2}{27} \text{ tanks per minute} \]

### Step 6: Determine how long it takes for pipe C to empty the tank
The rate of pipe C is \(\frac{2}{27}\) tanks per minute, which means it can empty the tank in:
\[ \text{Time for C} = \frac{1 \text{ tank}}{\frac{2}{27} \text{ tanks per minute}} = \frac{27}{2} \text{ minutes} = 13.5 \text{ minutes} \]

### Final Answer
Pipe C alone can empty the full tank in **13.5 minutes**.