Understanding Stanley's Water Tank Problem A Mathematical Analysis

Let's delve into a fascinating mathematical problem involving Stanley and his endeavor to fill a water tank. This scenario presents a practical application of linear functions, providing an excellent opportunity to understand how mathematical models can represent real-world situations. In this detailed exploration, we will dissect the problem, analyze the given function, and draw meaningful conclusions about the water tank's filling process. We will also look into similar concepts to help you better grasp the core ideas and tackle similar problems confidently.

Understanding the Initial Conditions

The problem states that Stanley is filling a water tank that already contains 10 gallons of water. This is a crucial piece of information because it establishes the initial condition of the problem. In mathematical terms, this represents the y-intercept of our linear function, which is the amount of water in the tank at the very beginning (when time, or the number of minutes, is zero). The initial 10 gallons act as the starting point, the foundation upon which the rest of the filling process is built. It's like having a head start in a race – Stanley isn't starting from an empty tank; he's already got a significant amount of water inside.

Understanding this initial condition is paramount because it directly impacts the overall amount of water in the tank at any given time. If the tank had started empty, the calculations would be different. The 10-gallon head start significantly alters the final outcome and must be considered when analyzing the problem. Furthermore, this initial value will be a critical component in constructing the linear equation that models the water tank's filling process. It serves as the constant term in the equation, representing the fixed amount of water present regardless of how many minutes pass.

Analyzing the Filling Rate

The problem further states that Stanley is filling the tank at a rate of 4.75 gallons per minute. This is another crucial piece of information, representing the rate of change or the slope of our linear function. The rate of 4.75 gallons per minute tells us how much the water level increases for every minute that passes. It's a constant rate, meaning that Stanley is adding the same amount of water to the tank each minute. This constant rate is what makes the relationship linear, meaning it can be represented by a straight line on a graph. The slope, 4.75, dictates the steepness of this line – a steeper slope indicates a faster filling rate, while a gentler slope would indicate a slower rate.

This filling rate is essential for calculating the total amount of water in the tank after a specific number of minutes. By multiplying the rate by the number of minutes, we can determine how many gallons have been added to the tank. However, we must not forget the initial 10 gallons already present. The total amount of water will be the sum of the initial amount and the amount added over time. This highlights the importance of both the initial condition and the rate of change in determining the overall behavior of the system. The filling rate is not merely a number; it's a dynamic factor that dictates how the water level progresses over time, making it a central element in understanding the entire scenario.

The Function Representing the Situation

The problem mentions that the situation is represented by a function, where $n$ is the number of minutes. This function, though not explicitly stated in the prompt, is the key to unlocking the solution. Based on the information provided, we can deduce the function's structure. Since we have an initial amount of water (10 gallons) and a constant filling rate (4.75 gallons per minute), we can express the total amount of water in the tank, which we'll call $W(n)$, as a function of the number of minutes, $n$. The function will take the form of a linear equation:

$$W(n) = 4.75n + 10$$

In this equation:

• W(n)$ represents the total amount of water in the tank (in gallons) after $n$ minutes.

• 4.75$ is the rate of filling (in gallons per minute), which is the slope of the line.

• n$ is the independent variable, representing the number of minutes that have passed.

• 10$ is the initial amount of water in the tank (in gallons), which is the y-intercept of the line.

This function serves as a mathematical model of the real-world situation. It allows us to predict the amount of water in the tank at any given time simply by plugging in the number of minutes. For instance, if we want to know how much water is in the tank after 10 minutes, we can substitute $n = 10$ into the function:

$$W(10) = 4.75(10) + 10 = 47.5 + 10 = 57.5$$

This calculation tells us that after 10 minutes, there will be 57.5 gallons of water in the tank. The function, therefore, provides a powerful tool for understanding and predicting the behavior of the system.

Exploring Linear Functions and Their Applications

This water tank problem is a classic example of a linear function in action. Linear functions are a fundamental concept in mathematics, and they have countless applications in various fields, including physics, economics, and computer science. A linear function is a function whose graph is a straight line. It can be represented in the general form:

$$y = mx + b$$

Where:

• y$ is the dependent variable.

• x$ is the independent variable.

• m$ is the slope, representing the rate of change.

• b$ is the y-intercept, representing the initial value.

In our water tank problem, $W(n)$ corresponds to $y$, $n$ corresponds to $x$, $4.75$ corresponds to $m$, and $10$ corresponds to $b$. Understanding this general form allows us to recognize and analyze linear relationships in various contexts. For example, if we were analyzing the cost of a taxi ride, the total cost might be a linear function of the distance traveled, where the slope is the cost per mile and the y-intercept is the initial fare.

Linear functions are particularly useful because they are easy to understand and work with. They have a constant rate of change, making predictions and calculations straightforward. However, it's important to remember that not all real-world situations can be accurately modeled by linear functions. Some relationships are non-linear, meaning their rate of change is not constant. In such cases, more complex functions may be needed to represent the situation accurately.

Solving Related Problems

To solidify your understanding of this concept, let's consider some related problems. Suppose Stanley had a different initial amount of water, say 5 gallons, and was filling the tank at a different rate, say 6 gallons per minute. We could then create a new function to represent this situation:

$$W(n) = 6n + 5$$

Now, we could ask various questions about this new scenario, such as:

• How much water will be in the tank after 15 minutes?
• How long will it take for the tank to contain 100 gallons?

To answer the first question, we simply substitute $n = 15$ into the function:

$$W(15) = 6(15) + 5 = 90 + 5 = 95$$

So, after 15 minutes, there will be 95 gallons of water in the tank.

To answer the second question, we need to solve the equation for $n$ when $W(n) = 100$:

$$100 = 6n + 5$$

Subtract 5 from both sides:

$$95 = 6n$$

Divide both sides by 6:

$$n = 95 / 6 ≈ 15.83$$

So, it will take approximately 15.83 minutes for the tank to contain 100 gallons. These types of problems reinforce the application of linear functions and the ability to manipulate equations to solve for unknown variables. By practicing with different scenarios and variations, you can develop a deeper understanding of the underlying mathematical principles.

Conclusion

Stanley's water tank problem provides a valuable illustration of how linear functions can be used to model real-world situations. By understanding the initial conditions, the rate of change, and the general form of a linear equation, we can effectively analyze and predict the behavior of the system. This problem also highlights the importance of translating word problems into mathematical expressions and using those expressions to solve for unknown quantities. The ability to work with linear functions is a crucial skill in mathematics and has wide-ranging applications in various fields. As you continue your mathematical journey, remember that practice and exploration are key to mastering these concepts and applying them to new and challenging problems.

• Linear functions: The mathematical concept underlying the problem, involving a constant rate of change.
• Rate of change: The 4.75 gallons per minute, representing how quickly the tank is filling.
• Initial condition: The 10 gallons of water already in the tank, the starting point for the filling process.
• Mathematical model: The function $W(n) = 4.75n + 10$, representing the relationship between time and the amount of water.
• Solving equations: The process of finding the value of an unknown variable, such as determining how long it takes to reach a certain water level.
• Word problems: Translating real-world scenarios into mathematical expressions.
• Slope: The steepness of the line representing the linear function, equivalent to the rate of change.
• Y-intercept: The point where the line intersects the y-axis, representing the initial value.
• Independent variable: The number of minutes, $n$, which can be varied.
• Dependent variable: The total amount of water, $W(n)$, which depends on the number of minutes.
• Function notation: Expressing the relationship between variables using function notation, such as $W(n)$.
• Predictions: Using the mathematical model to estimate future values, such as the amount of water after a certain time.
• Applications of linear functions: Real-world scenarios where linear functions can be used, such as taxi fares or filling containers.
• Graphing linear functions: Visualizing the relationship between variables by plotting the line on a graph.
• Constant rate: A rate of change that remains the same over time, a key characteristic of linear functions.
• Variables: Symbols representing unknown quantities, such as $n$ for the number of minutes.
• Equations: Mathematical statements showing the equality between two expressions.
• Problem-solving: The process of applying mathematical concepts and skills to solve real-world problems.