Water Tub Problem: Quarts Remaining Over Time

Hey guys! Ever wondered how quickly a tub empties? Let's dive into a classic problem where we explore the relationship between time and the amount of water left in a tub. We'll break it down step by step, making it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Problem

In this water tub problem, we have a tub that starts with 50 quarts of water. The tub empties at a constant rate of 2.5 quarts per minute. We want to understand how the amount of water left in the tub changes over time. To do this, we'll use variables: 'w' to represent the quarts of water remaining and 't' to represent the time in minutes. This is a classic scenario that helps us visualize linear relationships, which are fundamental in mathematics and real-life applications. Understanding the rate at which the tub empties is key to figuring out how much water is left at any given time. The problem gives us a starting point (50 quarts) and a rate of change (2.5 quarts per minute), which are the building blocks for our analysis. We'll see how these pieces fit together to form a clear picture of the water level in the tub over time. The goal here isn't just to solve a math problem, but to develop a deeper understanding of how quantities change in relation to each other, a skill that's useful in many areas of life.

Setting Up the Equation

To begin, let's translate the problem into a mathematical equation. We know the tub starts with 50 quarts, so that's our initial value. Since the water is draining, the amount decreases over time. The rate of decrease is 2.5 quarts per minute. Therefore, for every minute that passes, we subtract 2.5 quarts from the initial 50 quarts. We can express this relationship with a linear equation. The equation will take the form w = initial amount - (rate of decrease * time). In our case, this translates to w = 50 - 2.5t. This equation is crucial because it allows us to calculate the amount of water remaining ('w') at any given time ('t'). It’s a simple yet powerful way to model a real-world situation. Understanding how to set up such equations is a fundamental skill in algebra and helps in problem-solving across various contexts. We’re not just dealing with numbers here; we're creating a model that represents the physical process of water draining from a tub. This skill of translating word problems into mathematical expressions is essential for anyone looking to tackle more complex mathematical challenges.

Completing the Table

Now, let's use our equation to fill in the blanks in the table provided. We already have two data points: when time (t) is 0 minutes, the water remaining (w) is 50 quarts, and when time (t) is 2 minutes, the water remaining (w) is 45 quarts. To complete the table, we'll substitute different values of 't' into our equation w = 50 - 2.5t and calculate the corresponding 'w' values. This process helps us visualize how the water level changes over time. For example, if we want to know how much water is left after 4 minutes, we plug in t=4 into the equation. If we want to know the remaining water after 10 minutes, we plug in t=10. By systematically doing this for each time value in the table, we build a complete picture of the water draining process. This is a practical application of our equation, showing how it can be used to predict the amount of water remaining at any point in time. Remember, each calculation is a snapshot of the water level at a specific moment, and together, they tell the story of the tub emptying.

Let's calculate a few more data points:

• Time (t) = 4 minutes:
  • w = 50 - 2.5 * 4
  • w = 50 - 10
  • w = 40 quarts
• Time (t) = 6 minutes:
  • w = 50 - 2.5 * 6
  • w = 50 - 15
  • w = 35 quarts
• Time (t) = 8 minutes:
  • w = 50 - 2.5 * 8
  • w = 50 - 20
  • w = 30 quarts
• Time (t) = 10 minutes:
  • w = 50 - 2.5 * 10
  • w = 50 - 25
  • w = 25 quarts
• Time (t) = 12 minutes:
  • w = 50 - 2.5 * 12
  • w = 50 - 30
  • w = 20 quarts
• Time (t) = 14 minutes:
  • w = 50 - 2.5 * 14
  • w = 50 - 35
  • w = 15 quarts
• Time (t) = 16 minutes:
  • w = 50 - 2.5 * 16
  • w = 50 - 40
  • w = 10 quarts
• Time (t) = 18 minutes:
  • w = 50 - 2.5 * 18
  • w = 50 - 45
  • w = 5 quarts
• Time (t) = 20 minutes:
  • w = 50 - 2.5 * 20
  • w = 50 - 50
  • w = 0 quarts

Now we can complete the table:

Analyzing the Results

Looking at the completed table, we can see a clear pattern. For every 2 minutes that pass, the amount of water in the tub decreases by 5 quarts (2.5 quarts/minute * 2 minutes = 5 quarts). This consistent decrease is a hallmark of a linear relationship. The water level drops steadily until it reaches 0 quarts at 20 minutes. This analysis highlights the practical implications of our equation. We’re not just crunching numbers; we’re understanding how a real-world quantity changes over time. The table provides a visual representation of this change, making it easy to see the linear progression. Analyzing results like these helps us develop a stronger intuition for mathematical relationships and their applications in everyday situations. We can see when the tub will be half empty (at 10 minutes) or completely empty (at 20 minutes), which provides a concrete understanding of the rate of water drainage.

Real-World Applications

This water tub problem might seem simple, but it illustrates a fundamental concept that applies to many real-world situations. Linear relationships, like the one we've explored, are used to model a variety of phenomena, from calculating the distance traveled at a constant speed to predicting the decay of radioactive substances. Understanding these relationships helps us make informed decisions and predictions in our daily lives. For example, if you're filling a pool at a certain rate, you can use a similar equation to determine how long it will take to fill. Or, if you're tracking your spending, you can use a linear model to predict when you'll run out of funds. The ability to recognize and work with linear relationships is a valuable skill in many fields, including science, engineering, finance, and even everyday budgeting. By mastering the basics with problems like this, you’re building a foundation for understanding more complex models and applications in the future. So, next time you see a quantity changing at a constant rate, remember the water tub problem and how we used a simple equation to understand its behavior!

Conclusion

So, there you have it! We've successfully explored the relationship between the time and the amount of water remaining in a tub. We set up an equation, completed a table, analyzed the results, and even touched on real-world applications. Remember, understanding linear relationships is a valuable skill that can help you in many areas of life. Keep practicing, and you'll become a math whiz in no time! If you guys have any questions, feel free to ask. Happy problem-solving!