Question

A cistern has three pipes A, B, and C. Pipes A and B are inlet pipes, whereas C is an outlet pipe. Pipes A and B can fill the cistern separately in 3 hours and 4 hours respectively, while pipe C can empty the completely filled cistern in 1 hour. If the pipes A, B, and C are opened in order at 5, 6, and 7 a.m. respectively, at what time will the cistern be empty?

Hint:

For problems involving filling and emptying rates, calculate the net rate by adding inlet rates and subtracting outlet rates. Use the total volume to find the time required.

Answer 1

Step 1: Individual pipe rates per hour: Pipe A (inlet): \( \frac{1}{3} \) of cistern. Pipe B (inlet): \( \frac{1}{4} \) of cistern. Pipe C (outlet): \( \frac{1}{1} \) of cistern.
Step 2: From 5 a.m. to 6 a.m. (Pipe A open): Pipe A fills \( \frac{1}{3} \) of the cistern in 1 hour.
Step 3: From 6 a.m. to 7 a.m. (Pipes A and B open): The combined filling rate is \( \frac{1}{3} + \frac{1}{4} = \frac{7}{12} \) of the cistern per hour. In 1 hour, \( \frac{7}{12} \) of the cistern is filled.
Step 4: At 7 a.m. (Pipes A, B, and C open): The combined rate is \( \frac{1}{3} + \frac{1}{4} - 1 = \frac{-5}{12} \) of the cistern per hour. This indicates the cistern is emptying at \( \frac{5}{12} \) of its capacity per hour.
Step 5: Total filled by 7 a.m.: \( \frac{1}{3} \) (from 5-6 a.m.) + \( \frac{7}{12} \) (from 6-7 a.m.) = \( \frac{4}{12} + \frac{7}{12} = \frac{11}{12} \) of the cistern.
Step 6: Time to empty the cistern after 7 a.m.: The cistern is \( \frac{11}{12} \) full and emptying at \( \frac{5}{12} \) per hour. Time to empty = \( \frac{\frac{11}{12}}{\frac{5}{12}} = \frac{11}{5} \) hours, or 2.2 hours.
Step 7: Final time: 2.2 hours after 7 a.m. is 7:00 a.m. + 2 hours and 12 minutes = 9:12 a.m.
Final Answer: The cistern will be empty at \( \mathbf{9:12 \, {a.m.}} \).