Calculating The Rate Of Water Drainage In A Bathtub A Step-by-Step Guide
In this article, we delve into the rate of change in the amount of water draining from a bathtub over a specific period. We will analyze the given data, which presents the volume of water remaining in the bathtub at different time intervals, to determine how quickly the water is being drained. This involves calculating the rate of change, a fundamental concept in mathematics that describes how one quantity changes in relation to another. In our case, we'll be examining how the volume of water decreases as time progresses. This analysis will not only provide us with a numerical value for the rate of drainage but also offer insights into the real-world application of mathematical concepts. Understanding rates of change is crucial in various fields, from physics and engineering to economics and finance. By dissecting this seemingly simple scenario of a draining bathtub, we can grasp the underlying principles of this important mathematical tool and its relevance in everyday situations.
The table below illustrates the amount of water, measured in gallons, remaining in a bathtub as it is being drained over time, measured in minutes.
| Time (min) | Water (gal) |
|---|---|
| 7 | 38.5 |
| 8 | 36.25 |
| 9 | 34 |
| 10 | 31.75 |
Calculating the Rate of Change
To determine the rate of change, we need to understand the fundamental concept behind it. The rate of change represents how a dependent variable changes with respect to an independent variable. In simpler terms, it's the measure of how much one quantity changes for each unit change in another quantity. In the context of our bathtub example, the dependent variable is the amount of water (in gallons) remaining in the tub, and the independent variable is time (in minutes). We want to find out how many gallons of water are drained for each minute that passes. Mathematically, the rate of change is often represented as the slope of a line on a graph where the dependent variable is plotted against the independent variable. The slope, in turn, is calculated as the change in the dependent variable divided by the change in the independent variable. This is often expressed as rise over run, where the rise represents the change in the vertical axis (water volume in our case) and the run represents the change in the horizontal axis (time). By calculating the slope, we can precisely quantify the rate at which the water is draining from the bathtub, providing us with a clear understanding of the drainage process.
To calculate the rate of change, we can use the formula:
Rate of Change = (Change in Water) / (Change in Time)
We can pick any two points from the table to calculate the rate of change. Let's use the points (7, 38.5) and (8, 36.25).
Change in Water = 36.25 - 38.5 = -2.25 gallons Change in Time = 8 - 7 = 1 minute
Rate of Change = -2.25 gallons / 1 minute = -2.25 gallons per minute
Let's verify this by using another pair of points, (9, 34) and (10, 31.75):
Change in Water = 31.75 - 34 = -2.25 gallons Change in Time = 10 - 9 = 1 minute
Rate of Change = -2.25 gallons / 1 minute = -2.25 gallons per minute
As we can see, the rate of change is consistent across different pairs of points. This indicates that the water is draining at a constant rate. The negative sign signifies that the amount of water is decreasing over time. In other words, for every minute that passes, the bathtub loses 2.25 gallons of water. This consistent rate of drainage implies a linear relationship between the time elapsed and the amount of water remaining in the tub. The rate of change is a crucial parameter in understanding the dynamics of the drainage process, allowing us to predict how much water will be left in the bathtub at any given time. This concept of a constant rate of change is widely applicable in various real-world scenarios, such as calculating the speed of a moving object or the rate of population growth. Understanding the rate of change provides a powerful tool for analyzing and predicting trends in a wide range of phenomena.
Interpretation of the Rate of Change
The calculated rate of change of -2.25 gallons per minute holds significant meaning in understanding the drainage process of the bathtub. The negative sign is particularly important, as it indicates a decreasing trend. In this context, it signifies that the amount of water in the bathtub is diminishing as time progresses. The magnitude of the rate of change, 2.25 gallons, tells us the specific amount of water that is being drained from the tub every minute. This means that for each minute that elapses, 2.25 gallons of water are removed from the bathtub. This consistent rate of drainage suggests a steady and predictable process. We can confidently say that the drainage is not accelerating or decelerating significantly; it is occurring at a constant pace. This information can be valuable in various situations. For instance, if we know the initial amount of water in the tub, we can use the rate of change to estimate how long it will take for the tub to completely empty. Conversely, if we want to drain a specific amount of water, we can use the rate of change to calculate the required drainage time. This understanding of the rate of change allows us to effectively manage and predict the drainage process, making it a crucial piece of information for practical applications.
Real-World Applications of Rate of Change
The concept of rate of change, as demonstrated in the bathtub drainage example, extends far beyond simple household scenarios. It's a fundamental principle in various scientific, engineering, and economic fields. Understanding rates of change allows us to model and predict a wide range of phenomena. In physics, for instance, the rate of change of an object's position over time is its velocity, and the rate of change of velocity over time is its acceleration. These concepts are crucial for understanding motion and dynamics. In engineering, rates of change are used to design systems and processes, such as the flow of fluids in pipelines or the heat transfer in engines. Chemical reaction rates, which describe how quickly reactants are converted into products, are also examples of rates of change. In economics, rates of change are used to analyze economic growth, inflation, and interest rates. The growth rate of a country's GDP, the inflation rate (the rate of change of prices), and the interest rate (the cost of borrowing money) are all crucial indicators of economic health. In finance, rates of change are used to calculate investment returns, stock price fluctuations, and currency exchange rates. Understanding these rates of change is essential for making informed financial decisions. Even in everyday life, we encounter rates of change constantly. The speed at which we drive, the rate at which our phone battery drains, and the rate at which we consume food are all examples of rates of change that we deal with regularly. By understanding this fundamental concept, we can better interpret and interact with the world around us.
Conclusion
In conclusion, by analyzing the data provided, we have successfully determined the rate of change for the water draining from the bathtub. The calculated rate of change of -2.25 gallons per minute signifies a consistent and predictable drainage process. This means that for every minute that passes, the amount of water in the tub decreases by 2.25 gallons. The negative sign is crucial in indicating this decreasing trend. This exercise demonstrates the practical application of the rate of change concept, which is a fundamental principle in mathematics and various other fields. The ability to calculate and interpret rates of change allows us to understand and predict how quantities change in relation to one another. As we've discussed, this concept has wide-ranging applications in physics, engineering, economics, finance, and even everyday life. From calculating the speed of a moving object to analyzing economic growth rates, the rate of change is a powerful tool for understanding and modeling the world around us. This simple example of a draining bathtub serves as a clear illustration of this important mathematical concept and its real-world relevance. Understanding rates of change empowers us to make informed decisions and predictions in a variety of situations, making it an essential skill in today's world.
