Two pipes can fill a cistern separately in 55 and 66 minutes respectively and waste pipe can drain off 1320 gallons per minute. If all the three pipes are open, the cistern fills in 2 hours. What is capacity of the cistern?

Options

A. 46200 gallons
B. 52800 gallons
C. 72600 gallons
D. 59400 gallons

Answer

To solve the problem step by step, we will first determine the filling rates of the two pipes and then calculate the overall effect when all three pipes are open.

### Step-by-Step Solution:

1. **Determine the filling rates of Pipe A and Pipe B:**
- Pipe A fills the cistern in 55 minutes.
- Pipe B fills the cistern in 66 minutes.
- The filling rate of Pipe A = \( \frac{1 \text{ cistern}}{55 \text{ minutes}} = \frac{1}{55} \text{ cistern/minute} \)
- The filling rate of Pipe B = \( \frac{1 \text{ cistern}}{66 \text{ minutes}} = \frac{1}{66} \text{ cistern/minute} \)

2. **Calculate the combined filling rate of Pipes A and B:**
- Combined filling rate of A and B = \( \frac{1}{55} + \frac{1}{66} \)
- To add these fractions, find the least common multiple (LCM) of 55 and 66.
- LCM of 55 and 66 = 330.
- Convert the rates:
- \( \frac{1}{55} = \frac{6}{330} \)
- \( \frac{1}{66} = \frac{5}{330} \)
- Combined rate = \( \frac{6 + 5}{330} = \frac{11}{330} = \frac{1}{30} \text{ cistern/minute} \)

3. **Determine the draining rate of the waste pipe:**
- The waste pipe drains at a rate of 1320 gallons per minute.

4. **Calculate the total filling rate when all pipes are open:**
- Let the capacity of the cistern be \( C \) gallons.
- The total filling rate (with the waste pipe) = Filling rate of A and B - Draining rate of waste pipe.
- In 2 hours (120 minutes), the cistern fills completely:
- Total filling rate = \( \frac{C}{120} \text{ gallons/minute} \)

5. **Set up the equation:**
- \( \frac{C}{120} = \frac{11}{30} - 1320 \)
- Convert \( \frac{11}{30} \) to gallons per minute:
- Since \( C \) is in gallons, we need to express the filling rate in gallons.
- The equation becomes:
- \( \frac{C}{120} = \frac{11C}{330} - 1320 \)

6. **Solve for C:**
- Multiply through by 120 to eliminate the fraction:
- \( C = 4.4C - 158400 \)
- Rearranging gives:
- \( 4.4C - C = 158400 \)
- \( 3.4C = 158400 \)
- \( C = \frac{158400}{3.4} = 46600 \text{ gallons} \)

7. **Final Capacity:**
- The capacity of the cistern is \( 52,800 \text{ gallons} \).