Perimeter Of A Rectangle Expression And Calculation

Hey guys! Today, we're diving into the world of rectangles, expressions, and perimeters. We've got a fun problem where Zohar is using scissors to cut out a rectangle, and we need to figure out an expression for the perimeter and then calculate it when x has a specific value. Let's get started!

Understanding the Problem: Zohar's Rectangle

So, our friend Zohar is a bit of a crafting whiz and is cutting out a rectangle from a larger piece of paper. This rectangle has a length of 5x - 2 and a width of 3x + 1. Our mission, should we choose to accept it (and we do!), is twofold:

1. Find an expression that represents the perimeter of this rectangle.
2. Calculate the actual perimeter when x = 4.

Before we jump into the algebra, let's take a moment to refresh our memory about rectangles and perimeters. A rectangle, as we all know, is a four-sided shape with four right angles. The perimeter, in simple terms, is the total distance around the outside of the shape. Think of it as walking along all the edges of the rectangle – the total distance you walk is the perimeter.

For any rectangle, the perimeter can be calculated using the formula: Perimeter = 2 * (length + width). This makes sense because a rectangle has two sides of equal length (the lengths) and two sides of equal width (the widths). We're essentially adding up all four sides.

Now, let's apply this to Zohar's rectangle. We know the length is 5x - 2 and the width is 3x + 1. We can substitute these expressions into our perimeter formula. This is where the algebra fun begins!

Cracking the Code: Finding the Perimeter Expression

Okay, let's roll up our sleeves and get algebraic! We know the perimeter formula is Perimeter = 2 * (length + width). We also know that:

• Length = 5x - 2
• Width = 3x + 1

So, we can substitute these into the formula:

Perimeter = 2 * ((5x - 2) + (3x + 1))

Now, we need to simplify this expression. The first step is to combine the like terms inside the parentheses. Remember, like terms are terms that have the same variable raised to the same power (or no variable at all, which are called constants).

Inside the parentheses, we have 5x and 3x, which are like terms. We also have -2 and +1, which are also like terms (constants). Let's combine them:

Perimeter = 2 * (5x + 3x - 2 + 1)

Perimeter = 2 * (8x - 1)

Great! We've simplified the expression inside the parentheses. Now, we need to distribute the 2. This means we multiply the 2 by each term inside the parentheses:

Perimeter = 2 * 8x - 2 * 1

Perimeter = 16x - 2

Boom! We've done it! The expression that represents the perimeter of Zohar's rectangle is 16x - 2. This is a general formula that works for any value of x. It tells us that the perimeter is always 16 times the value of x, minus 2. But, we're not done yet. The next part of the mission is to find the actual perimeter when x = 4.

The Grand Finale: Calculating the Perimeter when x = 4

We've got our perimeter expression: Perimeter = 16x - 2. Now, we know that x = 4. So, all we need to do is substitute 4 for x in the expression:

Perimeter = 16 * 4 - 2

Time for some arithmetic! First, we multiply 16 by 4:

Perimeter = 64 - 2

Then, we subtract 2 from 64:

Perimeter = 62

And there you have it! When x = 4, the perimeter of Zohar's rectangle is 62 units. Remember, the units will depend on the units used for the length and width (e.g., inches, centimeters, etc.).

We've successfully cracked the code! We found the expression for the perimeter (16x - 2) and calculated the perimeter when x = 4 (which is 62). Give yourselves a pat on the back – you've earned it!

Wrapping Up: Why This Matters

This problem might seem like just a math exercise, but it actually highlights some important concepts that are used in many real-world situations. Understanding how to work with expressions and substitute values is crucial in fields like engineering, architecture, and even computer programming.

For example, imagine an architect designing a building. They need to calculate the perimeter of rooms to determine how much flooring or molding to order. They might use expressions like the ones we worked with today to account for different room sizes or variable dimensions.

So, by mastering these skills, you're not just learning math – you're building a foundation for solving real-world problems. Keep practicing, keep exploring, and keep those math muscles strong!

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Conclusion

We've successfully navigated the world of rectangles, expressions, and perimeters. From crafting the expression to calculating the final value, we've seen how algebra can be used to solve real-world problems. So, the next time you see a rectangle, remember Zohar and her scissors, and you'll be ready to tackle any perimeter challenge that comes your way!