A cistern has 3 pipes A and B and C,A and B can fill it in 2 and 3 hours respectivly,C is a waste pipe.If all the 3 pipes be opned at once,7/24 of the cistern wil be filled up in 30 minutes.In what time can C empty the full cistern ?

Options

A. 3 hours
B. 4 hours
C. 5 hours
D. 6 hours

Answer

To solve the problem step by step, we will analyze the information given and calculate the required time for pipe C to empty the full cistern.

### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 2 hours. Therefore, the rate of pipe A is:
\[ \text{Rate of A} = \frac{1}{2} \text{ cisterns per hour} \]
- Pipe B can fill the tank in 3 hours. Therefore, the rate of pipe B is:
\[ \text{Rate of B} = \frac{1}{3} \text{ cisterns per hour} \]

### Step 2: Calculate the combined filling rate of pipes A and B
- The combined rate of pipes A and B is:
\[ \text{Combined Rate of A and B} = \frac{1}{2} + \frac{1}{3} \]
- To add these fractions, we find a common denominator (which is 6):
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
- Thus,
\[ \text{Combined Rate of A and B} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \text{ cisterns per hour} \]

### Step 3: Determine the filling rate when all three pipes are opened
- It is given that when all three pipes (A, B, and C) are opened together, they fill \(\frac{7}{24}\) of the cistern in 30 minutes (which is \(\frac{1}{2}\) hour).
- Therefore, the rate of filling when all three pipes are opened is:
\[ \text{Rate of A, B, and C} = \frac{7}{24} \div \frac{1}{2} = \frac{7}{12} \text{ cisterns per hour} \]

### Step 4: Set up the equation to find the rate of pipe C
- The rate of pipe C is negative since it is a waste pipe. Let the rate of pipe C be \(-\frac{1}{x}\) cisterns per hour, where \(x\) is the time in hours it takes for C to empty the full cistern.
- The equation for the rates is:
\[ \frac{5}{6} - \frac{1}{x} = \frac{7}{12} \]

### Step 5: Solve for \(x\)
- Rearranging the equation gives:
\[ \frac{5}{6} - \frac{7}{12} = \frac{1}{x} \]
- To subtract \(\frac{7}{12}\) from \(\frac{5}{6}\), convert \(\frac{5}{6}\) to twelfths:
\[ \frac{5}{6} = \frac{10}{12} \]
- Now we can subtract:
\[ \frac{10}{12} - \frac{7}{12} = \frac{3}{12} = \frac{1}{4} \]
- Therefore, we have:
\[ \frac{1}{x} = \frac{1}{4} \]
- Taking the reciprocal gives:
\[ x = 4 \]

### Conclusion
- Pipe C can empty the full cistern in **4 hours**.