Margot's Pillow Perimeter Project: Finding The Difference In Ribbon Length

Margot, a passionate seamstress, is embarking on a new sewing project: crafting decorative pillows with ribbon trim. Her first pillow is a classic square, and she's carefully sewing a ribbon along its entire perimeter. The side length of this pillow is given by the algebraic expression 2x^2 + 1 inches, where 'x' is a variable that determines the size. Margot's creative mind is already envisioning her next project: a similar square pillow, also adorned with ribbon. However, this time, the side length is described by a different expression: 4x - 7 inches. Our task is to determine an expression that represents the difference in the amount of ribbon Margot needs for these two pillows.

To fully appreciate the nuances of this problem, we must delve into the fundamental concept of a square's perimeter. The perimeter, in essence, is the total distance around the shape. For a square, this is simply the sum of the lengths of all four sides. Since all sides of a square are equal in length, the perimeter is calculated by multiplying the side length by 4. This foundational understanding is the key to unlocking the solution to Margot's ribbon dilemma. We will explore how to apply this concept to the algebraic expressions representing the side lengths of Margot's pillows, ultimately leading us to the desired expression for the ribbon difference.

Before diving into the calculations, let's take a moment to consider the significance of algebraic expressions in this context. These expressions, with their variables and constants, allow us to represent a range of possible pillow sizes. The variable 'x' acts as a placeholder, and by assigning different numerical values to 'x', we can generate different side lengths and, consequently, different perimeters. This flexibility is a powerful tool in design and mathematics, allowing for generalization and exploration of various possibilities. In Margot's case, the expressions 2x^2 + 1 and 4x - 7 define the dimensions of her pillows, and our challenge is to compare the ribbon requirements for these algebraically defined squares.

To solve Margot's ribbon conundrum, we must first determine the perimeter of each pillow. Remember, the perimeter of a square is found by multiplying the side length by 4.

For the first pillow, the side length is 2x^2 + 1 inches. Therefore, the perimeter, which represents the length of ribbon needed, is 4 * (2x^2 + 1). To simplify this expression, we distribute the 4 across the terms inside the parentheses: 4 * 2x^2 + 4 * 1, which results in 8x^2 + 4 inches. This expression tells us the total length of ribbon Margot needs for the first pillow, dependent on the value of 'x'. The 8x^2 term signifies that the ribbon length grows quadratically with 'x', meaning the ribbon requirement increases at an accelerating rate as 'x' increases. The constant term, +4, represents the base ribbon requirement even when 'x' is zero.

Now, let's turn our attention to the second pillow. Its side length is given by 4x - 7 inches. Applying the same principle, we multiply the side length by 4 to find the perimeter: 4 * (4x - 7). Distributing the 4, we get 4 * 4x - 4 * 7, which simplifies to 16x - 28 inches. This linear expression (where the highest power of 'x' is 1) indicates that the ribbon length for the second pillow increases proportionally with 'x'. The coefficient 16 tells us the rate of increase, and the constant term -28 represents a subtraction from the ribbon length, suggesting that for smaller values of 'x', the second pillow might require less ribbon than the first.

With the perimeters of both pillows calculated, we've effectively quantified the ribbon needed for each. The expression 8x^2 + 4 inches represents the ribbon requirement for the first pillow, and 16x - 28 inches represents the ribbon requirement for the second. However, our original problem asked for the difference in ribbon lengths. To find this, we must perform one final step: subtracting the perimeter of the second pillow from the perimeter of the first pillow. This subtraction will reveal the expression that directly represents the additional ribbon Margot needs for the first pillow compared to the second. The next section will guide us through this crucial subtraction process, leading us to the final answer.

We've successfully determined the perimeters of Margot's two pillows: 8x^2 + 4 inches for the first pillow and 16x - 28 inches for the second. Now, to find the difference in ribbon lengths, we need to subtract the perimeter of the second pillow from the perimeter of the first. This can be represented mathematically as: (8x^2 + 4) - (16x - 28).

It's crucial to pay close attention to the order of subtraction and the signs when dealing with algebraic expressions. Subtracting an entire expression requires us to distribute the negative sign across each term within the parentheses. So, the expression becomes: 8x^2 + 4 - 16x + 28. Notice how the -16x term remains negative, but the -28 term becomes +28 due to the subtraction.

Now, we need to combine like terms to simplify the expression. Like terms are those that have the same variable raised to the same power. In this case, we have the quadratic term 8x^2, the linear term -16x, and the constant terms +4 and +28. Combining the constant terms, we get 4 + 28 = 32. Therefore, the simplified expression representing the difference in ribbon lengths is: 8x^2 - 16x + 32 inches.

This quadratic expression is the key to understanding the ribbon difference between Margot's pillows. The 8x^2 term indicates that the difference grows rapidly as 'x' increases, similar to the perimeter of the first pillow. The -16x term suggests that for smaller values of 'x', the difference might be smaller, and the +32 term represents a constant addition to the difference. However, the overall behavior of the expression is dominated by the quadratic term, meaning that for larger 'x' values, the first pillow will require significantly more ribbon than the second.

The expression 8x^2 - 16x + 32 inches is more than just a mathematical result; it provides valuable insight into Margot's sewing project. This expression directly answers the question of how much more ribbon Margot needs for the first pillow compared to the second, depending on the value of 'x'. By substituting different numerical values for 'x', Margot can determine the precise ribbon difference for pillows of various sizes.

For instance, if Margot decides that 'x' is equal to 3, she can substitute this value into the expression: 8(3)^2 - 16(3) + 32. This simplifies to 8(9) - 48 + 32, which further simplifies to 72 - 48 + 32, resulting in 56 inches. This means that when 'x' is 3, Margot needs 56 inches more ribbon for the first pillow than the second. This type of calculation allows Margot to plan her ribbon purchases and manage her materials effectively.

Furthermore, the expression can help Margot understand the relationship between pillow size and ribbon consumption. The quadratic nature of the expression, with the 8x^2 term, reveals that the ribbon difference increases at an accelerating rate as 'x' grows. This implies that for larger pillows, the first pillow will require significantly more ribbon than the second. This knowledge can inform Margot's design choices and help her optimize her material usage. She might, for example, decide to prioritize making the second type of pillow (with side length 4x - 7) if she wants to minimize ribbon consumption, especially for larger sizes.

In conclusion, by calculating the perimeters of the two pillows and finding the difference, we've not only solved the mathematical problem but also gained a practical understanding of Margot's sewing project. The expression 8x^2 - 16x + 32 inches serves as a powerful tool for Margot, allowing her to plan her ribbon usage, compare ribbon requirements for different pillow sizes, and ultimately, bring her creative vision to life with precision and efficiency. The blend of mathematical principles and practical application underscores the beauty and relevance of algebra in everyday scenarios.