There are 3 taps, A, B and C, which can fill a tank in 20, 30 and 40 minutes, respectively. The tank has 2 holes, one at the bottom of the tank, and the other at half the height of the tank. The hole at the bottom of the tank can empty it in 120 minutes and the other hole also empties at the same rate. If all the taps are opened, find the time taken to fill the tank. min min min min
Question
There are 3 taps, A, B and C, which can fill a tank in 20, 30 and 40 minutes, respectively. The tank has 2 holes, one at the bottom of the tank, and the other at half the height of the tank. The hole at the bottom of the tank can empty it in 120 minutes and the other hole also empties at the same rate. If all the taps are opened, find the time taken to fill the tank. min min min min
Solution
To solve this problem, we first need to find the rate at which each tap and hole fills or empties the tank.
- Tap A fills the tank in 20 minutes, so its rate is 1/20 tanks per minute.
- Tap B fills the tank in 30 minutes, so its rate is 1/30 tanks per minute.
- Tap C fills the tank in 40 minutes, so its rate is 1/40 tanks per minute.
- The hole at the bottom empties the tank in 120 minutes, so its rate is -1/120 tanks per minute (negative because it's emptying the tank).
- The hole at half the height also empties the tank in 120 minutes, so its rate is -1/120 tanks per minute.
If all the taps and holes are opened, the combined rate is the sum of the individual rates:
Combined rate = (1/20) + (1/30) + (1/40) - (1/120) - (1/120) = 0.05 + 0.0333 + 0.025 - 0.0083 - 0.0083 = 0.0917 tanks per minute.
To find the time taken to fill the tank, we divide 1 (the size of the tank) by the combined rate:
Time = 1 / 0.0917 = 10.9 minutes.
So, it will take approximately 11 minutes to fill the tank when all the taps and holes are opened.
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