There is a leak at the bottom of a cistern. Due to this it takes 8 hours to fill the cistern. Had there not be

Question

There is a leak at the bottom of a cistern. Due to this it takes 8 hours to fill the cistern. Had there not been a leak, it would have taken one hour less to fill the cistern. How much time does it take for the leak to completely empty the cistern?

Examoogle · Accepted Answer

The correct answer is Option A: 56 hours. Use the formula $T_{leak} = \frac{T_1 	imes T_2}{T_2 - T_1}$ where $T_1$ is the time with the leak (8 hours) and $T_2$ is the time without the leak (7 hours). Thus, $T_{leak} = \frac{8 	imes 7}{8 - 7} = 56$ hours.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the formula $T_{leak} = \frac{T_1 	imes T_2}{T_2 - T_1}$ where $T_1$ is the time with the leak (8 hours) and $T_2$ is the time without the leak (7 hours). Thus, $T_{leak} = \frac{8 	imes 7}{8 - 7} = 56$ hours.

Formula Used: $$\frac{1}{T_{net}} = \frac{1}{T_{fill}} - \frac{1}{T_{leak}}$$

Given: Time taken to fill the cistern with leak = 8 hours. Time taken without the leak = 8 - 1 = 7 hours.

Steps: Define Rates: Let the inlet rate be $I$ and the leak rate be $L$. The time taken to fill without the leak is 7 hours, so $I = 1/7$ tank/hour. With the leak, the time taken is 8 hours, so $I - L = 1/8$ tank/hour.. Calculate Leak Rate: Substitute $I$ into the second equation: $1/7 - L = 1/8$. Rearranging gives $L = 1/7 - 1/8$.. Final Result: The time required to empty the cistern is the reciprocal of the leak rate: $1 / (1/56) = 56$.

Answer: A is correct because the leak reduces the net flow rate such that the effective rate of emptying is $1/56$ per hour, resulting in $56$ hours to empty.

Exam Tip: This is essentially a 'Work and Time' problem in disguise. Remember that leak rate is treated as negative work!

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