Water Volume Increase Problem Solving A Mathematical Exploration

by Axel Sørensen 65 views

Hey guys! Let's dive into a fun mathematical problem about water volume increase. This problem involves a container initially holding 6 liters of water, with an additional 0.7 liters being poured in every hour. We're going to explore how to express this situation mathematically, calculate the total volume after a certain time, and consider the container's capacity. Get ready to put on your math hats and let's get started!

Problem Breakdown

Before we jump into solving the problem, let's break it down into smaller, manageable parts. This will help us understand the problem better and make it easier to tackle. The problem presents us with a scenario involving a container that initially has 6 liters of water. Water is being added to this container at a rate of 0.7 liters per hour. Our mission, should we choose to accept it, is threefold:

  • Express this information using mathematical language.
  • Calculate the amount of water in the container after 8.5 hours.
  • Determine when the container will reach its capacity.

Sounds like a plan? Awesome! Let's get started with the first part: expressing the information mathematically.

a. Expressing the Information Mathematically

Okay, so the first part of our mission is to translate the problem's description into a mathematical expression. This might sound intimidating, but trust me, it's like learning a new language – once you grasp the basics, it becomes super easy! The key here is to identify the variables and relationships involved. We know the initial volume of water is 6 liters, and the volume increases by 0.7 liters every hour. Let's use 'V' to represent the total volume of water in the container (in liters) and 't' to represent the time elapsed (in hours). We need to formulate an equation that connects these variables.

Think of it this way: the total volume, V, will be the initial volume plus the added volume. The added volume is simply the rate of water being poured in (0.7 liters per hour) multiplied by the time elapsed (t hours). Therefore, we can express the relationship as:

V = 6 + 0.7t

This is our mathematical expression! It tells us how the total volume, V, changes with time, t. The equation is a linear equation, which makes sense because the water is being added at a constant rate. The number 6 represents the initial amount of water, and the term 0.7t represents the additional water added over time. To make sure we're on the same page, let's break down this equation further. The '6' is a constant, meaning it doesn't change. It's the starting point. The '0.7' is the rate of change – how much the volume increases for each hour that passes. And 't' is our variable – it can be any number of hours, and the total volume will change accordingly. This equation is a powerful tool because it allows us to predict the volume of water in the container at any given time.

Now, let's put this equation to work and answer the next part of our problem.

b. Calculating the Water Volume After 8.5 Hours

Alright, now that we have our mathematical expression, we can use it to calculate the volume of water in the container after 8.5 hours. This is where things get really fun because we're applying our equation to a specific scenario. Remember our equation? It's V = 6 + 0.7t. We want to find V when t = 8.5 hours. So, all we need to do is substitute 8.5 for t in the equation and solve for V.

Let's do it step by step:

  • V = 6 + 0.7 * 8.5
  • V = 6 + 5.95
  • V = 11.95

So, after 8.5 hours, the container will have 11.95 liters of water. Isn't that neat? We used our mathematical model to predict a real-world outcome. To make this even clearer, let's think about what this result means in the context of the problem. We started with 6 liters, and over 8.5 hours, we added 0.7 liters each hour. Our calculation shows that the total volume after this time is 11.95 liters. This makes sense because it's more than the initial 6 liters, and it reflects the added water. Now, let's try to visualize this. Imagine a container with markings for each liter. Initially, the water level is at the 6-liter mark. As time passes, the water level steadily rises, and after 8.5 hours, it reaches almost the 12-liter mark. This visualization can help solidify our understanding of the problem and the solution. We've successfully calculated the volume after a specific time. Now, let's tackle the final part of our problem: figuring out when the container will reach its capacity.

c. Determining the Container's Capacity

Here's the final piece of our water volume puzzle: determining when the container will reach its capacity. To answer this, we need some additional information – the container's maximum volume. The original problem states: "Si la capacidad del recipiente...". Let's assume, for the sake of this example, that the container's capacity is 15 liters. Of course, we can adapt this process for any capacity! This part of the problem is about figuring out how long it takes for the water level to reach the top of the container. We'll use our trusty equation again, V = 6 + 0.7t, but this time, we know V (the capacity) and need to solve for t (the time).

So, if the capacity is 15 liters, we set V = 15 and solve for t:

  • 15 = 6 + 0.7t

To solve for t, we need to isolate it on one side of the equation. First, we subtract 6 from both sides:

  • 15 - 6 = 0.7t
  • 9 = 0.7t

Now, we divide both sides by 0.7:

  • t = 9 / 0.7
  • t ≈ 12.86 hours

Therefore, it will take approximately 12.86 hours for the container to reach its capacity of 15 liters. Let's pause and think about what this result tells us. It means that if we continue pouring water into the container at a rate of 0.7 liters per hour, it will overflow after about 12.86 hours. This is a crucial piece of information, especially in real-world situations where overfilling a container could lead to spills or damage. To get a better sense of this, let's consider a slightly different scenario. What if the container had a capacity of 10 liters instead? How would that change our calculation? We would simply set V = 10 in our equation and solve for t. The process would be the same, but the final answer would be different, reflecting the smaller capacity. This highlights the flexibility of our mathematical model – it can be adapted to different situations by changing the values of the variables involved. We've successfully calculated the time it takes to reach the container's capacity. Now, let's wrap up our discussion and reflect on what we've learned.

Conclusion

We've successfully navigated through a water volume problem, expressing the information mathematically, calculating the volume after a specific time, and determining the container's capacity. By using a simple linear equation, we were able to model a real-world scenario and make predictions. This problem highlights the power of mathematics in understanding and solving everyday situations. Guys, I hope this journey has been enlightening and that you've gained a better appreciation for the role of math in our lives. Keep exploring, keep questioning, and keep those mathematical gears turning!