Understanding the Dynamics of Cistern Filling and Emptying: A Mathematical Approach

Introduction

The problem involving a cistern, a tap, and a waste pipe is a common example in fluid mechanics and rate problems. This article aims to explore the mathematical principles behind such problems and how to solve them. By analyzing the rates of filling and emptying, we can determine the conditions under which a cistern can be filled or emptied.

The Problem Statement

The scenario involves a cistern that is initially full. When a waste pipe is open, it takes 30 minutes to empty the cistern completely. If, however, a tap is opened to fill it, it takes 40 minutes to empty the cistern. We need to determine how long it will take to fill the cistern when both the tap and the waste pipe are open.

Analysis of the Problem

Let's denote the capacity of the cistern as V liters. We need to find the rates at which the cistern is being filled and emptied.

Rate of Emptying by the Waste Pipe

The waste pipe can empty the cistern in 30 minutes. Therefore, the rate of emptying by the waste pipe is:

Rate of emptying (frac{V}{30}) liters per minute.

Rate of Filling by the Tap

The tap, when opened, takes 40 minutes to drain the cistern. However, we know that the tap can only fill the cistern, so we need to consider the net effect when both the tap and the waste pipe are open.

Let's denote the rate of the tap as R_t. The net rate at which water is being added or subtracted when both the tap and the waste pipe are open can be calculated as follows:

Net rate (frac{R_t - V}{30}).

We also know that in 40 minutes, the waste pipe will drain (frac{4}{3}V) liters, which is equivalent to:

Net rate (frac{4V - 3V}{30}) (frac{V}{30}) (cdot) 40 (frac{4V}{120}).

Simplifying this, we get:

Net rate (frac{4V}{120}) (frac{V}{30}) (cdot) 40 (frac{V}{3}).

Thus, the tap fills (frac{V}{120}) liters per minute.

Now, we can calculate the combined rate when both the tap and the waste pipe are open:

Rate of filling (frac{V}{30}) - (frac{V}{120}) (frac{4V - V}{120}) (frac{3V}{120}) (frac{V}{40}).

The rate of filling when both the tap and the waste pipe are open is (frac{V}{120}).

Conclusion

By analyzing the rates, we can conclude that the cistern will never be filled when both the tap and the waste pipe are open because the waste pipe's rate of emptying is higher than the tap's rate of filling. The cistern will continue to empty rather than fill.

Key Takeaways:

The rate of emptying by the waste pipe (V/30 liters per minute). The rate of filling by the tap (V/120 liters per minute). The net rate when both pipes are open (V/40 liters per minute – but this is negative, indicating net emptying).

For more complex fluid dynamics and related problems, it's important to break down the problem into smaller, manageable parts and analyze the rates involved. Understanding these principles will not only help in solving similar problems but also in real-world applications like water management and hydraulic systems.