Cube Side Length Problem Solving A Mathematical Puzzle

Hey guys! Today, we're diving into a super cool math problem that involves cubes, rectangular prisms, and a bit of algebra. We're going to figure out the side length of a cube after it's been transformed into a rectangular prism. It might sound a bit complex, but trust me, we'll break it down step by step. We will use the given equation $2x^3 + 8x^2 = 450$ to solve for x, which represents the original side length of the cube. This problem beautifully combines geometry and algebra, showing how mathematical concepts intertwine. So, let's get started and unlock the secrets hidden within this cubic puzzle!

Imagine we have a cube, a perfect six-sided shape where all sides are equal. Let's say each side of this cube has a length of x inches. Now, things get interesting. We're going to change this cube into a rectangular prism by altering its dimensions. One side is increased by 4 inches, and another side is doubled. This transformation changes the shape and, of course, the volume. We're told that the volume of this new rectangular prism is 450 cubic inches. And here's the key: we have an equation, $2x^3 + 8x^2 = 450$, that we can use to find the original side length x. This equation is our roadmap to solving the problem. The main goal here is to determine the value of x, which represents the original side length of the cube before any modifications were made. We'll explore how this equation is derived from the geometric properties of the cube and the rectangular prism, and then we'll solve it to find our answer. Understanding the problem statement is crucial because it sets the stage for the entire solution process. We need to visualize the cube, the changes made to it, and how these changes affect the volume. With a clear understanding of what we're trying to find, we can confidently move forward and tackle the equation.

Alright, let's break down how we got to the equation $2x^3 + 8x^2 = 450$. This is a crucial step in solving our problem. We started with a cube, right? Each side has a length of x. So, the volume of this cube would simply be x * x * x, which we write as $x^3$. Now, we're messing with the cube. One side gets 4 inches longer, so it becomes x + 4 inches. Another side is doubled, making it 2x inches. The third side remains the same at x inches. Remember, the volume of a rectangular prism is found by multiplying its length, width, and height. So, the volume of our new rectangular prism is (x + 4) * (2x) * (x). This simplifies to 2$x^2$(x + 4), which further expands to 2$x^3$ + 8$x^2$. We know from the problem statement that this new volume is 450 cubic inches. That's how we arrive at the equation 2$x^3$ + 8$x^2$ = 450. This equation is a mathematical representation of the geometric transformation we performed on the cube. It links the original side length x to the final volume of the rectangular prism. Setting up the equation correctly is super important because it's the foundation for finding the solution. If we mess up here, the rest of the process will be off. So, let's make sure we've got this straight before we move on to solving for x.

Okay, guys, we've got our equation: $2x^3 + 8x^2 = 450$. Now, let's roll up our sleeves and solve for x. The goal here is to isolate x and figure out its value. First things first, we want to simplify the equation a bit. Notice that all the terms have a common factor of 2. Let's divide both sides of the equation by 2. This gives us $x^3 + 4x^2 = 225$. Now, this is a cubic equation, which might seem intimidating, but don't worry, we'll tackle it. To solve it, we want to get all terms on one side and set the equation equal to zero. So, we subtract 225 from both sides, resulting in $x^3 + 4x^2 - 225 = 0$. Solving cubic equations can sometimes involve tricky methods, but in this case, we can try to find a simple integer solution by testing factors of 225. We're looking for a number that, when plugged in for x, makes the equation true. Let's try a few values. If we try x = 1, the equation doesn't hold. How about x = 2? Nope. Let's jump to x = 5. Plugging in 5, we get $5^3 + 4(5^2) - 225 = 125 + 100 - 225 = 0$. Bingo! So, x = 5 is a solution. Now that we've found one solution, we could use polynomial division or synthetic division to find the remaining solutions, but since we're dealing with a real-world problem (the side length of a cube), we're mainly interested in the positive real solution. In this case, x = 5 is likely the only solution that makes sense in our context. Therefore, we've successfully solved the equation and found that x = 5. This means the original side length of the cube was 5 inches. Solving the equation is the heart of the problem, and it requires careful algebraic manipulation and a bit of trial and error. But with a systematic approach, we can crack it!

Awesome! We've solved the equation and found that x = 5. But what does this really mean? Remember, x represents the original side length of our cube. So, we now know that the cube initially had sides that were 5 inches long. This is a fantastic discovery because it answers the main question posed in the problem. We started with a cube, changed its dimensions, and used an equation to backtrack and find its original size. The beauty of this problem is how it connects abstract algebra to a concrete geometric situation. We didn't just solve an equation; we solved a puzzle about a shape and its transformations. To recap, the original cube had sides of 5 inches each. One side was increased by 4 inches (becoming 9 inches), and another side was doubled (becoming 10 inches). The third side remained at 5 inches. These changes resulted in a rectangular prism with a volume of 450 cubic inches, which confirms our solution. Finding the side length is the culmination of our mathematical journey. It's the destination we were aiming for when we first encountered the problem. And now, with a clear understanding of the steps we took, we can confidently say that we've solved it.

Alright, guys, we did it! We successfully found the side length of the cube. We started with a geometric puzzle, translated it into an algebraic equation, and then solved that equation to find our answer. The original side length of the cube was 5 inches. This problem was a fantastic example of how math can be used to describe and solve real-world situations. We combined our knowledge of geometry (cubes, rectangular prisms, volume) with our skills in algebra (equation solving) to reach our goal. More importantly, we learned the value of breaking down a complex problem into smaller, manageable steps. We set up the equation, simplified it, and then systematically solved for x. This approach is super useful not just in math but in all sorts of problem-solving situations. So, what's the big takeaway here? Math isn't just about numbers and symbols; it's about thinking logically and creatively. It's about taking a challenge head-on and working through it step by step. And most importantly, it's about the satisfaction of finding the solution. We hope you enjoyed this mathematical adventure as much as we did. Keep exploring, keep questioning, and keep solving! Math is all around us, and it's waiting to be discovered. Until next time, keep those brains buzzing!