Solving Tank Filling Problems: A Detailed Guide

Understanding the dynamics of how different pipes fill up a tank is crucial in many real-world scenarios, such as water systems, manufacturing processes, and more. This guide will walk you through a step-by-step solution to a common problem involving pipes A and B filling a tank at different rates, and introduces the method of solving similar problems using the rates of individual pipes.

Problem Setup

Let's consider a tank that can be filled by pipe A in 3 hours and by pipe B in 6 hours. At 10 A.M., pipe A starts filling the tank. Pipe B starts at 11 A.M. We need to determine at what time the tank will be fully filled.

Understanding the Rates

First, we need to understand the individual rates of the pipes.

Rate of pipe A: Pipe A can fill the tank in 3 hours, thus it fills at a rate of 1/3 of the tank per hour.

Rate of pipe B: Pipe B can fill the tank in 6 hours, thus it fills at a rate of 1/6 of the tank per hour.

Combined Effect of Pipes

When both pipes are working together, their rates add up.

Combined rate: 1/3 1/6 2/6 1/6 3/6 1/2 of the tank per hour.

This means that working together, the pipes can fill half the tank in one hour.

Step-by-Step Solution

Step 1: Determine the Amount Filled by Pipe A Alone

From 10 A.M. to 11 A.M., pipe A works alone. Therefore, in one hour, pipe A will fill:

Amount filled by pipe A: 1/3 of the tank.

Step 2: Calculate the Remaining Tank

After one hour, the remaining amount of the tank to be filled is:

Remaining tank: 1 - 1/3 2/3 of the tank.

Step 3: Determine the Time to Fill the Remaining Tank

Pipe B starts at 11 A.M. Together, pipes A and B work at a rate of 1/2 of the tank per hour. We need to find out how long it will take for them to fill the remaining 2/3 of the tank:

Time required: (2/3) / (1/2) (2/3) × 2 4/3 hours.

Cutting 4/3 hours into minutes gives us: (4/3) × 60 minutes 80 minutes or 1 hour and 20 minutes.

Step 4: Calculate the Final Time

Starting from 11 A.M., adding 1 hour and 20 minutes provides the final time:

Final time: 11:00 A.M. 1:20 12:20 P.M.

General Solution Method

This problem can be generalized for any similar situation with multiple pipes with different filling rates. The key steps are to:

Determine the individual rates of each pipe. Find the combined rate. Calculate the amount filled by each pipe individually and together. Predict the final time based on the combined rate and the remaining amount of the tank.

For example, if pipe X fills a tank in 2 hours and pipe Y in 6 hours:

Step 1: Determine Individual Rates

Rate of pipe X: 1/2 of the tank per hour.

Rate of pipe Y: 1/6 of the tank per hour.

Step 2: Find the Combined Rate

Combined rate: 1/2 1/6 3/6 1/6 4/6 2/3 of the tank per hour.

Step 3: General Solution

If pipe X starts at 10 A.M. and pipe Y starts one hour later, the steps remain the same:

Pipe X fills 1/2 of the tank in one hour. The remaining tank is 1 - 1/2 1/2. Together, the pipes fill 2/3 of the tank in one hour, so the remaining 1/2 takes (1/2) / (2/3) (1/2) × (3/2) 3/4 hours or 45 minutes. Thus, the tank is filled at 11:45 A.M.

Conclusion

By following this guide, you can solve similar pipe filling problems effectively. Understanding the individual and combined rates of the pipes is key to determining the time required to fill a tank. The methods outlined here provide a clear and logical approach to solving such problems efficiently.