Calculating Cistern Leak Time
This problem involves calculating the time taken by a leak to empty a cistern, given the filling times of two taps and their combined filling time with the leak present.
Step 1: Determine Individual Tap Rates
- Rate of Tap 1 (filling): \(\frac{1}{8}\) cistern per minute.
- Rate of Tap 2 (filling): \(\frac{1}{20}\) cistern per minute.
Step 2: Calculate Combined Filling Rate (Without Leak)
If both taps worked together without any leak, their combined rate would be:
\( \text{Combined Rate (No Leak)} = \frac{1}{8} + \frac{1}{20} \)Find a common denominator (40):
\( = \frac{5}{40} + \frac{2}{40} = \frac{7}{40} \text{ cistern per minute} \)Step 3: Determine Actual Combined Rate (With Leak)
The problem states that together, the taps take 30 minutes to fill the cistern. This is the actual rate including the leak's effect:
\( \text{Actual Combined Rate (With Leak)} = \frac{1}{30} \text{ cistern per minute} \)Step 4: Calculate the Leak Rate
The leak empties the cistern, so its rate subtracts from the filling rate. The difference between the expected combined rate (without leak) and the actual combined rate (with leak) gives the leak's emptying rate:
\( \text{Leak Rate} = \left( \text{Combined Rate (No Leak)} \right) - \left( \text{Actual Combined Rate (With Leak)} \right) \) \( = \frac{7}{40} - \frac{1}{30} \)Find a common denominator (120):
\( = \frac{21}{120} - \frac{4}{120} = \frac{17}{120} \text{ cistern per minute} \)Step 5: Calculate Time for Leak to Empty Cistern
The time taken by the leak to empty a full cistern is the inverse of its rate:
\( \text{Time to Empty} = \frac{\text{Total Volume (1 Cistern)}}{\text{Leak Rate}} \) \( = \frac{1}{\frac{17}{120}} \text{ minutes} \) \( = \frac{120}{17} \text{ minutes} \)Similar Questions
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