Bigger Cube Thickness: 8 Cubes Melted Solution

Hey guys! Ever wondered what happens when you melt a bunch of smaller cubes and join them to make a bigger one? It's a cool concept, especially when we're talking about math and geometry! Let's break down a classic problem: imagine you have 8 metallic cubical blocks, all the same size. You melt them down and join them together to form one big cubical block. Now, if each of those smaller blocks is 10 cm thick, how do you figure out the thickness of the bigger block? Don't worry, we'll go through it step by step, making it super easy to understand. So, grab your thinking caps, and let's dive in!

Understanding the Problem

Before we jump into the solution, let's make sure we fully grasp what the problem is asking. We have eight identical metallic cubes. Each of these smaller cubes has a thickness (which is essentially the length of one of its sides) of 10 cm. The key here is that when we melt these cubes, the total volume of the metal doesn't change. It's just reshaped. This is a fundamental concept in physics and math: the conservation of volume.

Think of it like this: you have eight Lego bricks, all perfect cubes. If you melt them down and mold the melted plastic into one giant cube, the amount of plastic remains the same. Only the shape changes. Our goal is to find out the side length (or thickness) of this new, larger cube.

To solve this, we'll need to use some basic geometry, specifically the formula for the volume of a cube. We'll first calculate the volume of one small cube, then find the total volume of all eight cubes. Finally, we'll use that total volume to determine the side length of the larger cube. Ready? Let's get started with the first step: finding the volume of a single small cube.

Step 1: Calculate the Volume of a Smaller Cube

The volume of a cube is found by multiplying its side length by itself three times. In other words, it's side * side * side, or side³. Since each smaller cube has a thickness of 10 cm, the side length is 10 cm.

So, the volume of one small cube is:

Volume = side³ = 10 cm * 10 cm * 10 cm = 1000 cubic centimeters (cm³)

This means each of those eight little cubes takes up 1000 cm³ of space. Now that we know the volume of one small cube, the next step is to figure out the total volume of all eight cubes combined. This will give us the volume of the big cube we're trying to create. It's like figuring out how much total Lego plastic we have before molding it into the giant cube. Easy peasy, right? Let's move on to the next step!

Step 2: Calculate the Total Volume of Eight Cubes

Now that we know the volume of one small cube (1000 cm³), finding the total volume of all eight cubes is super straightforward. We simply multiply the volume of one cube by the number of cubes, which is 8.

Total Volume = Volume of one cube * Number of cubes

Total Volume = 1000 cm³ * 8 = 8000 cm³

So, the total volume of the metal we have to work with is 8000 cm³. This is the same amount of space the larger cube will occupy. Think of it as the total space taken up by all the Lego bricks, whether they are separate or melted into one big block. This total volume is the key to unlocking the final step: finding the thickness (side length) of the bigger cube. We're almost there, guys! Just one more step to go, and we'll have our answer. Let's jump into finding the thickness of the bigger cube!

Step 3: Determine the Thickness of the Bigger Cube

This is where we put it all together! We know the total volume of the bigger cube is 8000 cm³. We also know that the volume of a cube is side³. To find the side length (which is the thickness in this case), we need to find the cube root of the total volume. The cube root of a number is the value that, when multiplied by itself three times, gives you the original number.

So, we need to find the cube root of 8000 cm³.

Thickness = ∛(Total Volume)

Thickness = ∛(8000 cm³)

You might recognize that 8000 is a perfect cube. It's 20 * 20 * 20. So, the cube root of 8000 is 20.

Thickness = 20 cm

And there we have it! The thickness of the bigger cube is 20 cm. We've successfully navigated through the problem, step by step, and found our answer. Give yourselves a pat on the back! You've tackled a geometry problem like a pro. Now, let's recap the entire process to solidify our understanding.

Recap: Step-by-Step Solution

Let's quickly recap the steps we took to solve this problem. This will help reinforce the process and make sure we've got it down pat.

1. Calculate the Volume of a Smaller Cube: We started by finding the volume of one of the smaller cubes. Since each side was 10 cm, we calculated the volume as 10 cm * 10 cm * 10 cm = 1000 cm³.
2. Calculate the Total Volume of Eight Cubes: Next, we found the total volume of all eight cubes by multiplying the volume of one cube by 8. So, 1000 cm³ * 8 = 8000 cm³.
3. Determine the Thickness of the Bigger Cube: Finally, we found the thickness of the bigger cube by taking the cube root of the total volume. The cube root of 8000 cm³ is 20 cm.

So, the thickness of the bigger cube formed by melting and joining eight smaller cubes, each 10 cm thick, is 20 cm. See how breaking down the problem into smaller, manageable steps makes it so much easier? This approach can be applied to all sorts of math problems, and even challenges in everyday life.

Why This Matters: Real-World Applications

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