Analysis of Cistern Filling and Emptying Rates: A Comprehensive Guide

When dealing with systems that involve the filling and emptying rates of containers, one such scenario involves a cistern being filled by a pipe and emptied by a leak. In this article, we will explore several methods to determine the emptying time of a cistern due to a leak, using the given data and mathematical reasoning.

Method 1: Using Combined Rates and Efficiencies

Let us consider the first problem where a pipe can fill a cistern in 9 hours and a leak causes the cistern to fill in 10 hours. We can denote the rate at which the pipe fills the cistern as rp and the rate at which the leak empties the cistern as rl.

The rate at which the pipe fills the cistern, rp, is (frac{1}{6}) C/hour, assuming the capacity of the cistern is C and can be taken as 1 for simplicity. Therefore, the combined rate when both the pipe and the leak are in operation can be used to find the rate of the leak.

Using the equation 1 7rp - 7rl, we can solve for rl:

1 7((frac{1}{6})) - 7rl (-frac{1}{6}) -7rl (7rl frac{1}{6}) (rl frac{1}{42})

Thus, it would take 42 hours for the cistern to empty due to the leak.

Method 2: Using Time and Volume Relations

Another approach is to use the time and volume relations given in different scenarios. For instance, consider a situation where 10 hours of filling is compensated by 15 hours due to a leak. Here, 5 hours of filling is effectively canceled out by the leak in 15 hours. Therefore, 10 hours of filling would be canceled out in:

10/5 x 15 30 hours

This indicates that it takes 30 hours for the cistern to empty if it is full.

Method 3: Analyzing the Leakage and Filling Rates

In another method, it is stated that 2 hours of filling quantity gets leaked in 12 hours. Since 2 hours of filling is equivalent to 1/5 of the cistern's capacity, 1/5 of the cistern's capacity gets emptied in 12 hours. Therefore, the full tank will get emptied in:

12 x 5 60 hours

This method provides a straightforward calculation based on the relation between filling and leaking times.

Method 4: Using Combined Rate Analysis

A combined rate approach is used in another solution, where the filling rate of the first pipe is V/9, and the combined rate of filling and leaking is 1/12. The rate of the leak is derived from the equation 1/9 - 1/x 1/12, solving which gives:

1/x 1/9 - 1/12 1/36 Thus, it takes 36 hours for the leak to empty the cistern.

Method 5: Analyzing With Specific Time Relations

Another solution involves specific time relations. Given that the pipe alone can fill the cistern in 9 hours, and with the leak it takes 12 hours, the time taken by the leak to empty the cistern is calculated as:

12 x 9 / (12 - 9) 36 hours

This method provides a direct computation based on the given conditions.

In conclusion, the time taken by the leak to empty a full cistern can be calculated using various methods, ranging from combined rate analysis to specific time and volume relations. The most commonly derived result is 36 hours, indicating that the leak alone would take 36 hours to empty a full cistern.

Key Takeaways: - The combined rate of filling and leaking can be used to determine the individual rate of leaking. - Time relations between filling and leaking can provide a direct method to calculate the emptying time. - The specific conditions of filling and leaking times can be used to derive the exact emptying time.

Keywords: cistern filling, leak rate, emptying time, combined rate.