Consider three pipes P, Q and R. The pipes P and Q can fill the cistern in 2 and 2.5 hours respectively. The pipe R can empty the cistern in 1 hour. If these pipes were opened at an interval of half an hour at 6:00am, 6:30am and 7:00am respectively. When the cistern will get empty?
Solution
Correct Answer: Option D
Amount emptied by pipe R in one hour = 1
As pipe P fills the entire cistern in 2 hours,
Part of cistern filled by pipe P in one hour = ½
As pipe Q fills the entire cistern in 2.5 hours,
Amount filled by pipe Q in one hour = 1/2.5
For the first half an hour only pipe P is open.
∴ Part of cistern filled by pipe P in starting ½ hour = ½ × ½ = ¼
For the next half an hour, pipe P and pipe Q both are open.
∴ Part of cistern filled by pipe P and pipe Q in next half an hour \(= \frac{1}{2}\left( {\frac{1}{2} + \frac{2}{5}} \right) = \frac{1}{4} + \frac{1}{5} = \frac{9}{{20}}\)
∴ Total pat of cistern filled after one hour \(= \frac{1}{4} + \frac{9}{{20}} = \frac{7}{{10}}\)
After 1 hour all three pipes are open.
Let x be the number of hours for which all 3 pipes are open
∴ At the end of x hours, the cistern will be completely empty
\(\frac{7}{{10}} + x\left( {\frac{1}{2} + \frac{1}{{2.5}} - \frac{1}{1}} \right) = 0\)
\(\Rightarrow \frac{7}{{10}} - \frac{x}{{10}} = 0\)
⇒ x = 7
∴ All three tanks are open for 7 hours.
⇒ Tank will get empty 7 hours after 7:00 am.
⇒ Tank will get empty at 2:00 pm.
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