Cube Water Displacement: How Much Water Remains?

Hey guys! Ever wondered what happens when you drop a solid object into a container full of water? It's a classic problem involving volume and displacement, and today, we're diving deep into a cool mathematical scenario. We've got a cube filled with water, and we're about to drop a rectangular prism (a parallelepiped, to be precise) into it. The big question is: how much water will be left after this watery encounter? Let's break it down step by step, making sure we understand each concept along the way. We'll use some basic geometry and volume calculations, so get your thinking caps on! This is going to be a fun and insightful journey into the world of math, where we'll learn how to tackle real-world problems using our numerical skills. So, are you ready to become water displacement experts? Let's get started and see how much water remains in our cube!

Understanding the Problem

Okay, so here's the deal: We start with a cube, and this cube isn't just any cube – it's a big one! Each side, or edge, of this cube measures a whopping 50 centimeters. Now, imagine this cube filled to the brim with crystal-clear water. That's our starting point. Next, we introduce a parallelepiped. Don't let the fancy name scare you; it's basically a rectangular prism, like a brick or a box. This particular parallelepiped has sides measuring 30 cm, 30 cm, and 40 cm. Got the picture? Good. Now, the main question we need to answer is: When we carefully place this parallelepiped into the water-filled cube, how much water will be left inside the cube? To solve this, we need to understand a key concept: displacement. When we put the parallelepiped into the cube, it pushes some of the water out of the way. The amount of water that spills out is equal to the volume of the parallelepiped. So, to figure out how much water is left, we need to calculate the original volume of water in the cube and then subtract the volume of the parallelepiped. This problem brilliantly combines geometry and practical thinking, making it a fantastic exercise in problem-solving. We're not just crunching numbers; we're visualizing a real-world scenario and applying mathematical principles to find the answer. Let's dive into the calculations and uncover the solution!

Calculating the Volume of the Cube

Alright, let's get down to the nitty-gritty and start crunching some numbers! First up, we need to figure out the volume of the cube. Remember, our cube is a perfect shape where all sides are equal, and each edge measures 50 centimeters. Calculating the volume of a cube is super straightforward. It's simply the length of one side multiplied by itself three times. In mathematical terms, we say it's the side cubed, or side raised to the power of 3. So, in our case, the volume of the cube is 50 cm * 50 cm * 50 cm. Now, let's do the math: 50 multiplied by 50 is 2500, and then we multiply that by 50 again. What do we get? 125,000! So, the volume of our cube is 125,000 cubic centimeters (cm³). But wait, we're not quite done yet. The question asks for the remaining water in liters, not cubic centimeters. No sweat! We just need to do a quick conversion. Remember that 1 liter is equal to 1000 cubic centimeters. To convert our cubic centimeters to liters, we divide by 1000. So, 125,000 cm³ divided by 1000 gives us 125 liters. That's a pretty big cube of water! We now know that our cube initially holds 125 liters of water. This is a crucial piece of the puzzle, as it's the total amount of water we're starting with. Next, we'll calculate the volume of the parallelepiped, which will tell us how much water gets displaced. Stay with me, and let's keep this mathematical journey rolling!

Determining the Volume of the Parallelepiped

Okay, now that we've conquered the cube, let's turn our attention to the parallelepiped. As we discussed earlier, a parallelepiped is just a fancy name for a rectangular prism – think of it like a brick or a box. Our parallelepiped has dimensions of 30 cm, 30 cm, and 40 cm. Calculating the volume of a parallelepiped is also pretty simple. You just need to multiply its length, width, and height together. In our case, that's 30 cm * 30 cm * 40 cm. Let's break it down: First, multiply 30 by 30. That gives us 900. Now, we multiply 900 by 40. What's the result? 36,000! So, the volume of our parallelepiped is 36,000 cubic centimeters (cm³). Remember, this volume represents the amount of space the parallelepiped takes up when we drop it into the water. It's also the exact amount of water that will be displaced or spill out of the cube. Now, just like with the cube's volume, we need to express this in liters to keep our units consistent. We know that 1 liter is equal to 1000 cubic centimeters. So, to convert 36,000 cm³ to liters, we divide by 1000. 36,000 divided by 1000 is 36. Therefore, the volume of the parallelepiped is 36 liters. This means that when we place the parallelepiped into the cube, 36 liters of water will be displaced. We're getting closer to our final answer! We know the initial amount of water and the amount displaced. Next, we'll put it all together to find out how much water is left in the cube. Let's keep going!

Calculating the Remaining Water

Alright, guys, we're on the home stretch! We've done the heavy lifting of calculating the individual volumes, and now it's time for the grand finale: finding out how much water is left in the cube after we've displaced some with our parallelepiped. Remember, we started with a cube full of water, which we calculated to be 125 liters. Then, we introduced the parallelepiped, which displaced 36 liters of water. The logic here is straightforward: To find the remaining water, we simply subtract the volume of water displaced (the parallelepiped's volume) from the initial volume of water in the cube. So, our equation looks like this: Remaining Water = Initial Water Volume - Displaced Water Volume Plugging in our numbers, we get: Remaining Water = 125 liters - 36 liters Now, let's do the subtraction: 125 minus 36 equals 89. Therefore, the remaining water in the cube is 89 liters. And there you have it! We've successfully navigated the problem, calculated the necessary volumes, and arrived at our final answer. This problem perfectly illustrates how mathematical concepts can be applied to real-world scenarios. We didn't just crunch numbers; we visualized a physical situation and used our knowledge of geometry and volume to find the solution. Give yourselves a pat on the back – you've earned it! We've mastered the art of water displacement, and hopefully, you've gained a deeper understanding of how volume works. Math can be pretty cool, right?

Final Answer

So, let's wrap it all up with a nice, neat conclusion. After all our calculations and logical deductions, we've arrived at the final answer to our initial question: How much water will remain in the cube after we place the parallelepiped inside? We started with a cube that held 125 liters of water. We then introduced a parallelepiped that displaced 36 liters of water. By subtracting the displaced volume from the initial volume, we found that there are 89 liters of water remaining in the cube. Therefore, the final answer is 89 liters. This problem was a fantastic journey through the world of volumes and displacement, and we tackled it like pros! We not only found the numerical answer but also reinforced our understanding of how shapes, volumes, and real-world scenarios intertwine. Math isn't just about numbers; it's about understanding the world around us and solving problems in a logical and structured way. Keep practicing, keep exploring, and remember that every mathematical challenge is an opportunity to learn and grow. And with that, we've successfully solved another intriguing mathematical puzzle. Well done, everyone!