Leaking Tanks: Equations, Tables, And Water Loss

Hey there, math enthusiasts! Let's dive into a cool problem involving two water tanks, each losing water over time. We'll explore how equations and tables help us understand the water levels in these tanks. This is a classic example of linear equations in action, and it's super practical! We will see how these models help us visualize the water level decreasing in the tanks and how we can use the model to find different values such as how much water is in the tank after a certain amount of time or how long it takes to empty.

Tank A: Unveiling the Equation

Let's start with Tank A. We're given an equation that describes the amount of water in the tank, represented by y, as a function of time in minutes, denoted by x. The equation is: y = 900 - 11x. This equation is in the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. In this case, b (the y-intercept) is 900. This means that at the beginning (when x = 0 minutes), Tank A has 900 gallons of water. The slope, m, is -11. The slope is negative, which indicates that the water level is decreasing over time. More precisely, for every minute that passes, the water level in Tank A decreases by 11 gallons. The water is leaking out. Using the equation y = 900 - 11x, we can find how much water is in Tank A at any given time. For instance, after 10 minutes (x = 10), the water level would be y = 900 - 11*10 = 790 gallons. So, this equation provides us with a handy tool to predict the water level at any point in time.

Now, let's break down the significance of this equation. The term "-11x" represents the water lost over time. As x increases (time passes), the amount of water lost also increases. The coefficient -11 is the rate of change – the amount of water lost per minute. The starting value of the water level in the tank, which is 900 gallons, does not change. This helps us create a clear picture of how quickly the tank is emptying. The equation is a linear model, the equation is a straightforward way to understand the water loss, which makes the water level of the tank decrease at a constant rate over time. In a real-world scenario, this rate of leakage would likely be influenced by the size of the leak and the water pressure. This makes the equation so important for understanding the water level in the tank.

Let's also imagine some potential questions we could answer with this equation. How long will it take for Tank A to be empty? To find that out, we need to find the value of x when y equals zero, so we solve this equation: 0 = 900 - 11x. That helps us find at what time x, the water in the tank will be 0 gallons. Or maybe we want to know how much water will be left after a half-hour? We'd substitute 30 minutes for x. See how practical this is?

Tank B: Deciphering the Table

Now, let's shift our focus to Tank B. Unlike Tank A, we're given a table of values showing the water level in Tank B at different times. Let's take a look:

The table represents the amount of water in the tank, y, at different times, x. Initially, at x = 0 minutes, Tank B holds 985 gallons of water. As time progresses, the water level decreases. Notice the pattern: for every minute that passes, the water level decreases by 10 gallons. For example, from minute 0 to minute 1, the water level drops from 985 gallons to 975 gallons, which is a decrease of 10 gallons. Similarly, from minute 1 to minute 2, the water level decreases from 975 gallons to 965 gallons, again a decrease of 10 gallons. Thus, this pattern demonstrates a constant rate of change. This constant rate of change is also what makes the model linear, as the water level decreases at a constant rate over time.

We can determine the equation representing the water level in Tank B. The initial value (y-intercept) is 985, and the water level decreases by 10 gallons per minute (the slope). Therefore, the equation for Tank B is y = 985 - 10x. This equation enables us to calculate the water level at any point in time for Tank B. For instance, at 5 minutes, Tank B would have 985 - 10*5 = 935 gallons remaining. This equation works like the one for Tank A.

The table gives us key data points that we can use to develop the equation. The table is an important tool in the model to determine the amount of water over time. It makes it easy to visualize the water loss. Let’s consider other values. By observing the table, we understand how the water level changes over time. We could predict the water level at other times, or we could develop an equation for the water level in the tank. The table provides a practical means of tracking the water loss, and these data points help in predicting future water levels. Let’s imagine we want to know at what time the tank will be empty. The table gives us a way to track the values and create the equation. From that, we can also find the time when the tank is empty, we would use the equation. So, this table isn't just a set of numbers; it's a tool that helps us understand, predict, and ultimately solve problems related to the leaking water tank.

Comparing the Tanks: A Watery Showdown!

Alright, let's compare Tank A and Tank B. We know Tank A starts with 900 gallons and loses 11 gallons per minute, while Tank B begins with 985 gallons and loses 10 gallons per minute. This information allows us to directly compare the performance of each tank over time. We can compare the rate of water loss of each tank. Tank A is losing water faster (11 gallons per minute) than Tank B (10 gallons per minute). However, Tank B starts with more water, so it has a head start. If we want to know which tank will run out of water first, we would need to solve the equations: 0 = 900 - 11x for Tank A and 0 = 985 - 10x for Tank B to find out the time it will take for each tank to empty. This comparison helps illustrate the relationship between the initial value (the y-intercept) and the rate of change (the slope), demonstrating how these factors impact how quickly each tank empties. Knowing the equations, we could graph them on the same coordinate plane, allowing us to see visually how the water levels change over time and where the lines intersect, which would represent when both tanks have the same amount of water, although this will not happen with the provided equations. Comparing these equations provides a basis for understanding how different factors can impact the time it takes for a tank to empty. Therefore, this comparison provides valuable insights into the dynamics of water loss in each tank.

Imagine if we wanted to find out which tank has more water after a certain amount of time. Let's say we want to compare the water levels after 20 minutes. For Tank A: y = 900 - 11*20 = 680 gallons. For Tank B: y = 985 - 10*20 = 785 gallons. Tank B would have more water than Tank A at this time. This is also how we can use the equations and the tables in a practical way.

Conclusion: Mastering Leaky Tanks

So, there you have it, guys! We've explored two leaking water tanks, using equations and tables to understand how the water levels change over time. We found that the equation is a great way of describing the water loss in Tank A. We looked at how to find an equation using the table provided for Tank B, and then, using both equations, we could make comparisons. Both equations and tables are effective tools for modeling real-world situations and provide insights to predict the water level at any point in time. These concepts apply to many other real-life scenarios, from understanding the depreciation of an asset to calculating the trajectory of a projectile. Understanding linear equations is a fundamental skill in mathematics and provides a solid foundation for more advanced topics. By understanding linear equations, you can better model and understand many real-world phenomena.

Keep practicing, and you'll become a pro at solving these types of problems! I hope you enjoyed this journey into the world of leaking tanks and linear equations. Keep exploring, and you'll find math is full of interesting and useful applications. Happy problem-solving!