Rectangle Perimeter: Find The Expression
Hey guys! Today, we're diving into a fun geometry problem: figuring out the expression for the perimeter of a rectangle. Sounds tricky? Don't worry, we'll break it down together. We're given that the length of the rectangle is 3 units shorter than one-third of the width, which we'll call x. Our mission is to find the expression that represents the perimeter of this rectangle. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into the math, let's make sure we understand the basics. We need to really nail down the key components of this problem. What do we know? We know the width is x, and the length is related to x in a specific way. We also know that the perimeter of a rectangle is the total distance around it. This means we'll need to add up all the sides. Remember, a rectangle has two lengths and two widths. Let's put this into action. This is how we can apply the foundations of mathematics to unravel this geometry puzzle.
Defining the Variables
First things first, let's define our variables clearly. This will help us avoid confusion later on. We know:
- Width = x
- Length = (1/3)x - 3 (because it's 3 units shorter than one-third of the width)
Understanding these definitions is crucial. It's like laying the foundation for a building – if the foundation isn't solid, the whole structure can wobble! We've translated the word problem into mathematical expressions, and that's a huge step in the right direction. This is where the magic of algebra begins to unfold, turning words into equations that we can manipulate and solve.
Perimeter Formula
Next up, we need to remember the formula for the perimeter of a rectangle. It's pretty straightforward:
Perimeter = 2 * (Length + Width)
This formula is the key to unlocking our problem. It tells us exactly how to combine the length and width to find the total distance around the rectangle. Think of it as the recipe we need to follow to bake our mathematical cake – we have all the ingredients (length and width), and the formula tells us how to mix them together. Without this formula, we'd be wandering in the dark, so it's super important to have it locked in our memory banks.
Calculating the Perimeter
Now comes the fun part – plugging in our values and simplifying! We'll take the information we have and see what comes out in the end. Ready? Let's do it!
Substituting the Values
Let's substitute the expressions we defined earlier into the perimeter formula:
Perimeter = 2 * (((1/3)x - 3) + x)
This might look a little scary, but don't be intimidated! We're just replacing the words "Length" and "Width" with their mathematical equivalents. It's like swapping ingredients in a recipe – we're keeping the overall structure the same, but using different components. The key here is to be careful and methodical, making sure we substitute correctly and don't miss any terms. This step sets the stage for the simplification process, where we'll tidy things up and get to our final answer.
Simplifying the Expression
Now, let's simplify the expression step by step. Remember our order of operations (PEMDAS/BODMAS)? We'll start with the parentheses:
Perimeter = 2 * ((1/3)x + x - 3)
We need to combine the x terms. To do this, we can think of x as (3/3)x:
Perimeter = 2 * ((1/3)x + (3/3)x - 3)
Perimeter = 2 * ((4/3)x - 3)
Now, we distribute the 2:
Perimeter = 2 * (4/3)x - 2 * 3
Perimeter = (8/3)x - 6
And there we have it! The expression representing the perimeter of the rectangle is (8/3)x - 6. We took a complicated-looking formula and simplified it down to its most basic form. This is the beauty of algebra – we can manipulate expressions to make them easier to understand and work with. Each step in the simplification process is like peeling away a layer of an onion, revealing the core underneath. We've successfully navigated the math and arrived at our solution.
Identifying the Correct Option
Okay, we've got our expression: (8/3)x - 6. Now, let's compare it to the options given in the problem.
We were given the following options:
A. (2/3)x - 4 B. (8/3)x - 2 C. (2/3)x - 8 D. (8/3)x - 6
Matching the Expression
By comparing our simplified expression, (8/3)x - 6, to the options, we can see that it matches option D perfectly. Options A, B, and C have different coefficients for x or different constant terms, so they're not the right fit. Option D is our winner! This is the moment of truth, where all our hard work pays off. We've successfully translated the problem into mathematical language, simplified the expression, and identified the correct answer. It's like finding the missing piece of a puzzle – everything clicks into place, and we can see the complete picture.
Conclusion
So, the correct answer is D. (8/3)x - 6. We did it! We successfully found the expression for the perimeter of the rectangle. Remember, the key is to break down the problem into smaller, manageable steps. Define your variables, recall the relevant formulas, substitute carefully, and simplify step by step. With a little practice, you'll be a pro at these types of problems in no time!
Remember practice makes perfect. This is our chance to really solidify our understanding and build our problem-solving muscles. The more we practice, the more confident and comfortable we'll become with tackling challenging math problems. So, let's keep going, keep learning, and keep growing our mathematical skills! You guys rock!