Parallelepiped Volume Calculation: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem involving a rectangular parallelepiped. It sounds fancy, but don't worry, we'll break it down together. We're going to figure out how to calculate its volume when we know some key things about its edges and total surface area. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so here’s the deal. We have a rectangular parallelepiped, which is basically a fancy name for a box where all the angles are right angles. Think of a brick or a perfectly rectangular container. The problem tells us that the three edges that meet at one corner (or vertex) are three consecutive numbers. This is a crucial piece of information. It means if one edge is, say, x, then the next one is x + 1, and the one after that is x + 2. We also know the total surface area of this box is 94 square meters. Our mission, should we choose to accept it (and we do!), is to find the volume of this parallelepiped.

To really grasp what's going on, let's visualize this parallelepiped. Imagine holding a shoebox. The length, width, and height are those three consecutive numbers we talked about. The total surface area is the sum of the areas of all six sides of the box. And the volume, well, that’s how much space the box can hold. This initial visualization is critical. Picture those edges meeting at a vertex, each one a little longer than the last. Feel the surface area wrapping around the box, and imagine filling the box to find its volume. This mental image will be our guide as we move through the math.

Before we jump into the calculations, let's think about the formulas we'll need. The surface area of a rectangular parallelepiped is given by 2*(lw + lh + wh), where l is the length, w is the width, and h is the height. The volume, which is what we're trying to find, is simply lwh. See how these formulas relate to our visualization? The surface area is about adding up the areas of the faces, and the volume is about multiplying the three dimensions together. These are our tools, and now we need to use them to solve the puzzle. Remember, the key is to connect the math to the physical shape we’re imagining. By understanding the geometry, the algebra becomes much less intimidating. So, let’s keep that shoebox in mind as we move on to the next step: setting up our equations.

Setting Up the Equations

Alright, let's translate our word problem into some math equations. This is where the magic happens, guys! We know the three edges are consecutive numbers, so we can represent them as x, x + 1, and x + 2. These are our length, width, and height – but not necessarily in that order. It doesn’t matter which one is which for the volume calculation since multiplication is commutative (that’s a fancy word for saying the order doesn’t matter). But for the surface area, we need to make sure we account for all the pairs of dimensions.

The formula for the total surface area, as we discussed, is 2*(lw + lh + wh). We know this total surface area is 94 square meters. So, we can plug in our consecutive numbers for l, w, and h into this formula. Let’s say l = x, w = x + 1, and h = x + 2. Our equation then becomes: 2*[x(x + 1) + x(x + 2) + (x + 1)(x + 2)] = 94. Woah, that looks a bit intimidating, right? But don’t sweat it! We're going to simplify this step-by-step.

This equation is the heart of our problem. It links the unknown edge length, x, to the known surface area, 94. Our next step is to simplify this equation, which means expanding the terms inside the brackets and combining like terms. Think of it as tidying up a messy room – we're just organizing the math so we can see it more clearly. We'll be using the distributive property (remember that from algebra?) and carefully combining the x squared terms, the x terms, and the constants. It’s like putting together a puzzle – each piece needs to fit just right to reveal the final picture. And once we have that simplified equation, we’ll be one giant leap closer to finding the value of x, which will unlock the secrets of our parallelepiped’s dimensions and ultimately, its volume.

Solving for x

Okay, let’s tackle that equation we set up. Remember, it was 2*[x(x + 1) + x(x + 2) + (x + 1)(x + 2)] = 94. First, we need to expand the terms inside the brackets. Let’s take it one step at a time. x(x + 1) becomes x² + x. x(x + 2) becomes x² + 2x. And (x + 1)(x + 2) becomes x² + 3x + 2. See? We’re just using the distributive property, multiplying each term inside the parentheses by the term outside. It’s like unwrapping a present – each layer reveals something new.

Now, let’s substitute these expanded terms back into our equation: 2*[x² + x + x² + 2x + x² + 3x + 2] = 94. Next, we need to combine like terms inside the brackets. We have three x² terms, which combine to 3x². We have x, 2x, and 3x, which combine to 6x. And we have a constant term, 2. So, our equation simplifies to 2*[3x² + 6x + 2] = 94. We’re making progress! It’s like we’re zooming in on the answer, getting closer and closer with each step.

Now, let’s distribute the 2: 6x² + 12x + 4 = 94. To solve for x, we need to set the equation equal to zero. So, we subtract 94 from both sides: 6x² + 12x - 90 = 0. This is a quadratic equation, guys! Now, we can simplify this even further by dividing the entire equation by 6: x² + 2x - 15 = 0. Much better! This looks way more manageable. We’ve transformed a complex-looking equation into something we can actually work with. Now, we need to solve this quadratic equation for x. We can do this by factoring, using the quadratic formula, or completing the square. Factoring is often the easiest if we can find two numbers that multiply to -15 and add to 2. Can you think of what those numbers might be? Think about the factors of 15… 3 and 5, right? And to get a sum of 2, we need -3 and 5. So, we can factor our equation as (x - 3)(x + 5) = 0. This means that either x - 3 = 0 or x + 5 = 0. Solving for x in each case, we get x = 3 or x = -5. But wait a minute! Can an edge length be negative? Nope! So, we discard the solution x = -5. This means the value of x is 3.

Calculating the Volume

Alright, we've found our x! We know that x = 3. Remember, x represents one of the edges of our rectangular parallelepiped. The other edges are x + 1 and x + 2. So, the edges are 3, 4, and 5 meters. We are almost at the finish line, guys!

Now, to find the volume, we simply multiply these three dimensions together. The volume, V, is lwh, which is 3 * 4 * 5. Let’s do the math: 3 times 4 is 12, and 12 times 5 is 60. So, the volume of our rectangular parallelepiped is 60 cubic meters. We did it!

Let's take a moment to appreciate what we've accomplished. We started with a word problem that seemed a bit daunting. We visualized the shape, translated the words into equations, simplified a complex equation, solved for x, and finally, calculated the volume. Each step was a piece of the puzzle, and we put them all together to reveal the answer. This is the power of math – it allows us to solve real-world problems by breaking them down into smaller, manageable steps.

Final Answer

Therefore, the volume of the rectangular parallelepiped is 60 m³. So, the correct answer is (c) 60. Awesome job, everyone! You've successfully navigated this parallelepiped problem. Remember, math isn't about memorizing formulas; it's about understanding the concepts and applying them to solve problems. Keep practicing, keep exploring, and keep those brains working! And most importantly, have fun with it! You guys are rockstars!