Solving the Water Tank Problem: A Comprehensive Guide for SEO

Have you ever wondered how to solve a problem involving the filling and emptying rates of a water tank? This article provides a step-by-step solution to a common dilemma, breaking down the mathematics and explaining the underlying concepts in a way that is both accessible and SEO-friendly. By understanding the rates at which different taps fill and empty a tank, you can optimize your site's SEO and provide value to your audience.

Introduction

This article addresses a classic problem often found in mathematical and engineering contexts: determining how long it will take to fill a water tank when both a filling tap and an emptying tap are open. Let's delve into the problem and its solution, ensuring the content is optimized for Google's algorithms while maintaining clarity and practicality.

The Problem

The problem states that one tap fills a water tank in 12 minutes, while another tap empties the same tank in 15 minutes. When both taps are open, how long will it take to fill the tank?

Step-by-Step Solution

Filling and Emptying Rates

First, we need to determine the rates at which the tank is filled and emptied by the respective taps.

Filling Rate: Since one tap fills the tank in 12 minutes, the filling rate is frac{1}{12} of the tank per minute. Emptying Rate: Since another tap empties the tank in 15 minutes, the emptying rate is frac{1}{15} of the tank per minute.

Next, we calculate the net rate at which the tank is being filled when both taps are open.

Combined Rate

To find the net rate, we subtract the emptying rate from the filling rate:

frac{1}{12} - frac{1}{15}

Using a common denominator of 60, we can rewrite the rates as:

frac{5}{60} - frac{4}{60} frac{1}{60}

This means that the net rate at which the tank is being filled is frac{1}{60} of the tank per minute.

Time to Fill the Tank

Given the net rate of frac{1}{60} of the tank per minute, we can now determine the time it takes to fill the tank:

Time frac{1 tank}{frac{1}{60} tank/min} 60 minutes

Therefore, it will take 60 minutes to fill the water tank if both taps are left open.

Alternative Method

Let's consider a different scenario where the tank has a volume (X) gallons. The filling pipe delivers (X) gallons in 12 minutes, which is equivalent to (1/5) hour. Thus, the filling rate is (5X) gallons per hour. The emptying pipe drains (X) gallons in 15 minutes, which is also (1/4) hour, so the emptying rate is (5X) gallons per hour.

The net rate at which the tank is being filled is:

5X - 4X X gallons per hour

Given that the net rate is (X) gallons per hour, it will take 60 minutes (1 hour) to fill the tank.

Practical Application

In practical scenarios, such as managing water systems, understanding these rates is crucial. The concept of volume and rate of fill can be applied to various real-world problems, from plumbing systems to industrial processes.

Mathematically, we define the rate of fill as (frac{dV}{dt}), which can be visualized practically by measuring the gallons per minute or liters per minute. In this example, we used gallons per minute, which is a common measurement in the United States.

By adding the rates of the filling and emptying pipes, we can determine the net rate and then calculate the time needed to fill the tank. This method is not only useful for solving mathematical problems but also for optimizing real-world processes, ensuring efficiency and consistency.

Conclusion

Understanding the rates at which tanks are filled and emptied is a valuable concept with numerous practical applications. By following the step-by-step solution provided in this article, you can solve similar problems and apply the knowledge to various scenarios, whether in recreational or professional contexts. Optimizing your content for search engines, as demonstrated in this guide, can help you reach a broader audience and provide valuable information to your readers.