Water Tank Drainage Understanding Range In Mathematical Relations
Hey there, math enthusiasts! Let's dive into a super interesting problem involving a 500-gallon water tank that's draining at a rate of 50 gallons per minute. We're going to explore how the amount of water left in the tank depends on the number of minutes it has been draining, and most importantly, we'll figure out the range of this relation. This might sound a bit technical, but trust me, we'll break it down step by step so it's crystal clear. So, grab your thinking caps, and let's get started!
Understanding the Scenario
Before we jump into the nitty-gritty of the range, let's make sure we fully grasp the situation. We have a water tank that initially holds 500 gallons of water. Imagine it's like a giant bathtub filled to the brim. Now, this tank is draining water at a constant rate of 50 gallons every minute. Think of it as pulling the plug and watching the water swirl down the drain. The crucial thing to understand here is that the amount of water remaining in the tank is directly related to how long the tank has been draining. The longer it drains, the less water there will be. This relationship between the draining time and the remaining water is what we're going to analyze.
The relationship between the amount of water remaining and the draining time is key to solving this problem. The initial amount of water is 500 gallons, and it decreases as time passes. The rate of decrease is constant at 50 gallons per minute. This means that for every minute that goes by, the tank loses 50 gallons of water. This constant rate of change indicates a linear relationship. Linear relationships are the simplest to understand and work with, which is great news for us. We can express this relationship mathematically, which will help us determine the range.
To further illustrate this, let’s consider a few specific time points. At the start, when no time has passed (0 minutes), the tank is full with 500 gallons. After 1 minute, 50 gallons have drained, leaving 450 gallons. After 2 minutes, 100 gallons have drained, leaving 400 gallons, and so on. You can see the pattern here: for each additional minute, the water level drops by 50 gallons. This pattern continues until the tank is completely empty. Understanding this pattern is crucial for determining the range of the relation. The range, in this context, refers to all the possible values for the amount of water left in the tank. This is what we're aiming to find out, and it will give us a complete picture of how the water level changes over time.
Defining Range in Mathematical Relations
Okay, so we've got a good handle on the water tank scenario. But what exactly is the "range" we're trying to find? In the world of mathematical relations, the range is a fundamental concept. It refers to the set of all possible output values (often called the y-values) that a relation can produce. Think of it like this: if you have a machine that takes an input and spits out an output, the range is all the different outputs the machine can possibly give you.
In our water tank problem, the relation connects the time the tank has been draining (in minutes) to the amount of water left in the tank (in gallons). The time is our input, and the amount of water remaining is our output. So, the range we're looking for is the set of all possible amounts of water that can be left in the tank as it drains. To visualize this, imagine a graph where the x-axis represents time and the y-axis represents the amount of water. The range would be the portion of the y-axis that the graph covers. In simpler terms, it's the minimum and maximum water levels that are possible in this situation.
Understanding the range helps us to fully describe and interpret the relation. It tells us the boundaries within which the output values can exist. For instance, in our water tank example, the amount of water cannot be negative because you can't have less than no water in the tank. Similarly, the amount of water cannot exceed the initial capacity of the tank, which is 500 gallons. These boundaries are what we're trying to define when we determine the range. It’s like setting the limits for a game – you can’t score below zero, and there’s a maximum score you can achieve. The range does the same thing for mathematical relations.
Determining the Minimum and Maximum Water Levels
Now, let's get down to the nitty-gritty of finding the range for our water tank problem. To do this, we need to figure out the minimum and maximum amounts of water that can be in the tank. These two values will define the boundaries of our range. Let's start with the maximum water level. What's the most water the tank can hold? Well, that's easy – it's the initial amount, which is 500 gallons. This is our starting point, the very top of our range.
On the other end of the spectrum, what's the minimum amount of water that can be in the tank? Think about it: the tank will drain until it's completely empty. So, the minimum amount of water is 0 gallons. This is the bottom of our range. We've now identified the two extremes: 500 gallons when the tank is full and 0 gallons when it's empty. These values are the key to understanding the range of this relation.
However, it's not just about the endpoints. We also need to consider all the possible water levels in between 0 and 500 gallons. Since the water drains continuously and at a constant rate, the amount of water in the tank can be any value between these two extremes. It's like a smooth slide from 500 gallons down to 0 gallons. There are no jumps or gaps in the water level. This means that the range will include all the numbers between 0 and 500, which is a crucial detail when we express the range mathematically.
Calculating the Draining Time
Before we can nail down the range, there's another crucial piece of the puzzle we need to solve: how long does it take for the tank to drain completely? This will help us understand the time component of our relation and how it connects to the amount of water left in the tank. We know the tank starts with 500 gallons and drains at a rate of 50 gallons per minute. So, to find the total draining time, we can simply divide the initial amount of water by the draining rate. Think of it like dividing a total distance by a speed to find the time it takes to travel that distance.
In our case, we have 500 gallons / 50 gallons per minute = 10 minutes. This tells us that the tank will be completely empty after 10 minutes. This is a significant piece of information because it sets the boundary for the time variable in our relation. The tank can drain for a maximum of 10 minutes, and beyond that, there's no more water to drain. This time limit is directly related to the range we're trying to find. It tells us that the amount of water left in the tank is only meaningful within this 10-minute timeframe.
Understanding the draining time also helps us visualize the relationship between time and the amount of water. At 0 minutes, the tank is full. As time increases, the amount of water decreases until, at 10 minutes, the tank is empty. This linear decrease in water level over time is what defines the range of our relation. Knowing the draining time allows us to accurately represent this relationship and express the range in a clear and concise way.
Expressing the Range Mathematically
Alright, we've done all the groundwork, and now we're ready to express the range mathematically. This is where we put all our insights together into a neat and precise statement. Remember, the range is the set of all possible output values, which in our case is the amount of water left in the tank. We know the minimum amount of water is 0 gallons, and the maximum amount is 500 gallons. So, the amount of water can be any value between these two limits, including 0 and 500 themselves.
In mathematical notation, we can express this range using inequalities. We use the variable 'y' to represent the amount of water left in the tank (our output). The inequality will show that 'y' is greater than or equal to 0 (because we can't have negative water) and less than or equal to 500 (because that's the maximum capacity of the tank). This gives us the inequality 0 ≤ y ≤ 500. This is a concise and accurate way to describe the range of our relation.
But there's another important factor to consider: the time. We know the tank drains completely in 10 minutes. So, the amount of water in the tank is only relevant within this timeframe. This means we also need to consider the time component when expressing the range. The amount of water depends on how much time has passed since the draining started. So, while 0 ≤ y ≤ 500 correctly describes the possible amounts of water, we should also consider the time aspect to fully understand the relation. This is where the options given in the problem come into play, and we need to choose the one that best represents the range in the context of the draining time.
Evaluating the Given Options
Now, let's bring it all together and evaluate the given options to select the correct answer. We've determined that the range represents the possible amounts of water left in the tank, which can be any value between 0 and 500 gallons. We also know that the tank drains completely in 10 minutes, which means the time is a crucial factor in this relation.
Looking at the options, we need to choose the one that best reflects these conditions. Option A states "0 ≤ y ≤ 10". This inequality suggests that the range is between 0 and 10, which seems to relate to the time rather than the amount of water. However, it doesn't accurately represent the possible amounts of water in the tank, which can be up to 500 gallons. Therefore, Option A is not the correct answer.
Option B, which isn't provided in your initial query, would need to represent the range of the amount of water in the tank. To be correct, it should consider both the minimum (0 gallons) and the maximum (500 gallons) amounts of water, as well as the time it takes to drain. A correct option would likely express the range as 0 ≤ y ≤ 500, but it might also include a condition related to the time, such as stating that this range is valid for the time interval 0 to 10 minutes. Without the specific wording of Option B, it's impossible to definitively say if it's correct, but we can infer that it should accurately reflect the possible amounts of water within the relevant time frame.
Conclusion: The Correct Range
So, let's wrap things up and highlight the key takeaways from our exploration of the water tank problem. We've journeyed through understanding the scenario, defining the concept of range, determining the minimum and maximum water levels, calculating the draining time, and expressing the range mathematically. By breaking down each step, we've gained a comprehensive understanding of how to approach this type of problem.
The most important thing we've learned is that the range represents all the possible output values in a relation. In our case, it's the set of all possible amounts of water left in the tank. We found that the minimum amount is 0 gallons (when the tank is empty), and the maximum amount is 500 gallons (when the tank is full). We also realized that the draining time of 10 minutes plays a crucial role in defining the context of the range.
When evaluating the options, we saw that Option A, "0 ≤ y ≤ 10", didn't accurately represent the range because it focused on the time rather than the amount of water. To choose the correct answer, we need an option that reflects the possible water levels between 0 and 500 gallons. While we don't have Option B fully stated, we understand that it should express this range, possibly with a consideration for the 10-minute draining time. Understanding the range allows us to fully describe and interpret the relation between the draining time and the amount of water in the tank. Keep practicing these concepts, and you'll become a master of mathematical relations in no time! Great job, guys!