Cuboid Dimensions: Find Length From Volume & Ratios

Hey guys! Today, we're diving into a fun math problem involving cuboids. It's like a 3D rectangle, and we're going to figure out one of its sides using some cool ratios and its volume. So, let's break down this problem step by step. If you've ever wondered how dimensions relate to volume, or if you just love a good math puzzle, you're in the right place. This problem will help you understand how to apply ratios to solve real-world geometry problems. Grab your thinking caps, and let's get started!

Understanding the Problem

Okay, let's first make sure we really understand what we're dealing with. The keyword here is 'cuboid'. A cuboid, as you might know, is basically a 3D rectangle—think of a box. It has a length, a breadth (which is like its width), and a height. Now, the problem gives us some interesting relationships between these dimensions. Specifically, the breadth is twice the height, and the length is thrice the breadth. These relationships are crucial because they allow us to express all the dimensions in terms of a single variable, making the problem much easier to solve.

The problem also tells us the volume of this cuboid: 2592 cubic centimeters. Remember, volume is the amount of space a 3D object occupies, and for a cuboid, it's calculated by multiplying the length, breadth, and height. Our mission, should we choose to accept it (and we do!), is to find the length of this cuboid. Sounds like a plan? Great! Before we jump into the calculations, let's recap the key pieces of information. We know the relationships between the dimensions (breadth to height, length to breadth), we know the volume, and we're after the length. With these in mind, we can set up our equations and solve for the unknown. Let's get to the next section and start putting the pieces together!

Setting Up the Equations

Alright, now for the exciting part – turning words into math! This is where we translate the problem's descriptions into mathematical equations, which we can then solve. Remember, we have three dimensions to deal with: length, breadth, and height. Let’s use some symbols to represent them. We'll call the height 'h', since that seems like a natural choice. Now, here’s where the given relationships come into play.

The problem tells us the breadth is twice the height. So, if the height is 'h', then the breadth is simply 2 times 'h', or '2h'. Easy peasy, right? Next up, we know the length is thrice the breadth. Since we've already figured out the breadth is '2h', the length is 3 times '2h', which is '6h'. So, now we've expressed all three dimensions – height, breadth, and length – in terms of a single variable, 'h'. This is a crucial step because it simplifies our problem significantly. Instead of dealing with three unknowns, we only have one! Now, let's not forget the volume. We know the volume of the cuboid is 2592 cubic centimeters. And we also know that the volume of a cuboid is calculated by multiplying its length, breadth, and height. So, we can write another equation: length × breadth × height = volume. Substituting our expressions for length, breadth, and height, we get: (6h) × (2h) × (h) = 2592. There you have it! We've set up the main equation we need to solve for 'h'. Once we find 'h', we can easily calculate the length, which is what the problem asks for. So, are you ready to dive into solving this equation? Let's head to the next section and crack this math nut!

Solving for the Height

Okay, let's get down to business and solve for 'h'! We've got our equation: (6h) × (2h) × (h) = 2592. The first thing we want to do is simplify the left side of the equation. When you multiply these terms together, you're essentially multiplying the numbers and the 'h's separately. So, 6 times 2 is 12, and then we have h × h × h, which is h³. This gives us the simplified equation: 12h³ = 2592. Doesn't that look a bit more manageable? Now, our goal is to isolate 'h³'. To do that, we need to get rid of the 12 that's multiplying it. We can do this by dividing both sides of the equation by 12. Remember, whatever you do to one side of the equation, you have to do to the other to keep things balanced! So, 12h³ / 12 = 2592 / 12. This simplifies to h³ = 216. We're getting closer! We now know that 'h cubed' is 216. But we want to find 'h', not 'h³'. So, we need to undo that cubing. The opposite of cubing a number is finding its cube root. In other words, we need to find a number that, when multiplied by itself three times, equals 216. If you know your cubes, you might already know the answer. If not, don't worry! You can use a calculator or think about it a bit. What number times itself three times gives you 216? It's 6! So, the cube root of 216 is 6. Therefore, h = 6. Woohoo! We've found the height of the cuboid. But remember, the problem asks for the length, not the height. Don't worry; we're just one step away from the finish line. Let's head to the next section to calculate the length using the height we just found.

Calculating the Length

Alright, we've conquered the height; now, let's snag that length! We know from our earlier setup that the length is 6 times the height (length = 6h). And guess what? We just figured out that the height (h) is 6 centimeters. This makes finding the length super straightforward. All we need to do is substitute the value of 'h' into our equation for the length. So, length = 6 × 6. Do the math, and you get length = 36 centimeters. There you have it! We've found the length of the cuboid. Wasn't that a satisfying journey? We started with some relationships between the dimensions, used the volume to set up an equation, solved for the height, and finally, calculated the length. We've tackled this problem like true math detectives, piecing together the clues to reveal the answer. Now, before we wrap things up, let’s just take a quick moment to make sure our answer makes sense in the context of the problem. We found the length to be 36 cm. Since the breadth is twice the height (2 * 6 = 12 cm), and the length is thrice the breadth (3 * 12 = 36 cm), the relationships given in the problem hold true. Also, if we multiply the length, breadth, and height together (36 cm * 12 cm * 6 cm), we get 2592 cubic centimeters, which matches the given volume. This double-check helps us feel confident that our answer is correct. Now that we've confidently found the length, let's do a quick recap of our journey and see what we've learned along the way.

Final Answer and Recap

Okay, guys, let’s bring it all home! After our mathematical adventure, we've successfully found the length of the cuboid. The final answer is 36 centimeters. Give yourselves a pat on the back for sticking through the problem and cracking the code! Now, let's do a quick rewind and recap the steps we took to get here. First, we carefully read the problem and identified the key information: the relationships between the dimensions (breadth is twice the height, length is thrice the breadth) and the volume of the cuboid (2592 cubic centimeters). Then, we translated these relationships into algebraic equations. We represented the height as 'h', the breadth as '2h', and the length as '6h'. This was a crucial step because it allowed us to express all the dimensions in terms of a single variable. Next, we used the formula for the volume of a cuboid (length × breadth × height) to set up an equation involving 'h': (6h) × (2h) × (h) = 2592. We simplified this equation to 12h³ = 2592 and then solved for 'h'. We divided both sides by 12 to get h³ = 216 and then took the cube root of both sides to find h = 6 centimeters. Finally, we used the value of 'h' to calculate the length. Since length = 6h, we substituted h = 6 to get length = 36 centimeters. We also double-checked our answer to make sure it made sense in the context of the problem. Phew! We covered a lot of ground there. This problem wasn't just about finding a number; it was about understanding how dimensions relate to each other and to the volume of a 3D object. It’s about how to translate word problems into math problems, and about how to solve those problems step by step. I hope you enjoyed this math adventure as much as I did. Keep practicing, keep exploring, and you'll become math whizzes in no time! Until next time, happy problem-solving!