Two pipes, A and B, can fill a tank in 10 minutes and 20 minutes, respectively. The pipe C can empty the tank in 30 minutes. All the three pipes are opened at the same time in the beginning. However, pipe C is closed 2 minutes before the tank is filled. In what time will the tank be full (in minutes)?

Question

Accepted Answer

Let the total time to fill the tank be T minutes.

Rates:
- Pipe A fills the tank in 10 minutes, so rate of A = 1/10 tank/min.
- Pipe B fills the tank in 20 minutes, so rate of B = 1/20 tank/min.
- Pipe C empties the tank in 30 minutes, so rate of C = -1/30 tank/min (negative because it empties).

For the first (T - 2) minutes, all three pipes are open.
For the last 2 minutes, only pipes A and B are open.

Amount filled in first (T - 2) minutes:
= (rate of A + rate of B + rate of C) * (T - 2)
= (1/10 + 1/20 - 1/30) * (T - 2)

Calculate combined rate:
1/10 = 6/60
1/20 = 3/60
1/30 = 2/60
Sum = (6 + 3 - 2)/60 = 7/60 tank/min

Amount filled in last 2 minutes:
= (rate of A + rate of B) * 2
= (1/10 + 1/20) * 2
= (6/60 + 3/60) * 2 = (9/60) * 2 = 18/60 = 3/10 tank

Total amount filled = 1 tank

So,
(7/60)(T - 2) + 3/10 = 1

Multiply both sides by 60:
7(T - 2) + 18 = 60
7T - 14 + 18 = 60
7T + 4 = 60
7T = 56
T = 8 minutes

But this contradicts the options and the problem statement. Let's re-check calculations.

Recalculate combined rate:
1/10 = 0.1
1/20 = 0.05
1/30 = 0.0333
Sum = 0.1 + 0.05 - 0.0333 = 0.1167 tank/min

Amount filled in last 2 minutes:
= (0.1 + 0.05) * 2 = 0.15 * 2 = 0.3 tank

Total amount filled:
0.1167 * (T - 2) + 0.3 = 1
0.1167T - 0.2334 + 0.3 = 1
0.1167T + 0.0666 = 1
0.1167T = 0.9334
T = 0.9334 / 0.1167 ≈ 8 minutes

This suggests the tank fills in 8 minutes, but pipe C is closed 2 minutes before filling, so the total time is 8 minutes.

However, the problem states pipe C is closed 2 minutes before the tank is filled, so the total time is T, and pipe C is open for (T - 2) minutes.

Let's verify the answer options:
- 8 minutes (option 3) matches calculation.

Therefore, the correct answer is 8 minutes.

Answer

Answer

Answer