Question:
Two pipes A and B can fill a cistern in 24 hours and 30 hours respectively, while a waste pipe C can empty 35 liters of water per minute. If all three pipes are opened together when the cistern is empty, it gets filled completely in 40 hours. What is the capacity of the cistern?
दो नल A और B एक टंकी को क्रमशः 24 घंटे और 30 घंटे में भर सकते हैं, जबकि एक निकासी नल C प्रति मिनट 35 लीटर पानी बाहर निकालता है। यदि टंकी खाली होने पर तीनों नलों को एक साथ खोल दिया जाए, तो टंकी 40 घंटे में पूरी भर जाती है। टंकी की धारिता (क्षमता) क्या है?
Options:
- 42,000 liters
- 36,000 liters
- 48,000 liters
- 54,000 liters
- 42,000 लीटर
- 36,000 लीटर
- 48,000 लीटर
- 54,000 लीटर
Correct Answer: a)
Explanation:
Let capacity of the cistern be \(V\) liters.
In 1 hour:
Work of A = \(\frac{1}{24}\)
Work of B = \(\frac{1}{30}\)
Net work of (A + B – C) = \(\frac{1}{40}\)
Therefore, rate of emptying of pipe C in 1 hour:
\(\frac{1}{C} = \frac{1}{24} + \frac{1}{30} – \frac{1}{40}\).
LCM of 24, 30, 40 is 120.
\(\frac{1}{C} = \frac{5 + 4 – 3}{120} = \frac{6}{120} = \frac{1}{20}\).
Thus, pipe C alone can empty the full cistern in 20 hours.
In 20 hours, water emptied = \(20 \times 60 \text{ minutes} \times 35 \text{ liters/minute}\)
\(= 1200 \times 35 = 42,000\) liters.
Therefore, the capacity of the cistern is 42,000 liters.
In 1 hour:
Work of A = \(\frac{1}{24}\)
Work of B = \(\frac{1}{30}\)
Net work of (A + B – C) = \(\frac{1}{40}\)
Therefore, rate of emptying of pipe C in 1 hour:
\(\frac{1}{C} = \frac{1}{24} + \frac{1}{30} – \frac{1}{40}\).
LCM of 24, 30, 40 is 120.
\(\frac{1}{C} = \frac{5 + 4 – 3}{120} = \frac{6}{120} = \frac{1}{20}\).
Thus, pipe C alone can empty the full cistern in 20 hours.
In 20 hours, water emptied = \(20 \times 60 \text{ minutes} \times 35 \text{ liters/minute}\)
\(= 1200 \times 35 = 42,000\) liters.
Therefore, the capacity of the cistern is 42,000 liters.
1 घंटे में कार्य:
नल A = \(\frac{1}{24}\), नल B = \(\frac{1}{30}\), तीनों का कुल = \(\frac{1}{40}\)
निकासी नल C का प्रति घंटा कार्य:
\(\frac{1}{C} = \frac{1}{24} + \frac{1}{30} – \frac{1}{40} = \frac{5 + 4 – 3}{120} = \frac{6}{120} = \frac{1}{20}\)
अतः नल C अकेले 20 घंटे में टंकी खाली कर सकता है।
टंकी की क्षमता = \(20 \times 60 \times 35 = 42,000\) लीटर।
नल A = \(\frac{1}{24}\), नल B = \(\frac{1}{30}\), तीनों का कुल = \(\frac{1}{40}\)
निकासी नल C का प्रति घंटा कार्य:
\(\frac{1}{C} = \frac{1}{24} + \frac{1}{30} – \frac{1}{40} = \frac{5 + 4 – 3}{120} = \frac{6}{120} = \frac{1}{20}\)
अतः नल C अकेले 20 घंटे में टंकी खाली कर सकता है।
टंकी की क्षमता = \(20 \times 60 \times 35 = 42,000\) लीटर।