Pillow Perimeter: Comparing Ribbon Lengths

by Alex Johnson 43 views

Let's dive into a fun mathematical problem involving pillows and ribbons! This problem focuses on calculating and comparing the perimeters of two square pillows, which will help us determine the amount of ribbon needed to sew around their edges. We'll explore how to use algebraic expressions to represent the side lengths of these pillows and then calculate their perimeters. Understanding these concepts is crucial not only for solving mathematical problems but also for practical applications in everyday life, such as sewing projects or home improvement tasks. So, grab your thinking caps, and let's get started on this exciting journey of mathematical discovery!

Understanding the Problem

To begin, let’s clearly define the problem. Margot is sewing a ribbon around the perimeter of two square pillows. The first pillow has a side length of 2x² + 1 inches, while the second pillow has a side length of 4x - 7 inches. Our main goal is to determine and compare the lengths of ribbon Margot will need for each pillow. This involves calculating the perimeter of each square, which is the total length of all its sides. Since a square has four equal sides, the perimeter is simply four times the length of one side.

Key Concepts:

  • Perimeter of a Square: The perimeter of a square is found by the formula P = 4s, where P represents the perimeter and s represents the side length.
  • Algebraic Expressions: We are given the side lengths as algebraic expressions (2x² + 1 and 4x - 7). This means we will need to use algebraic manipulation to find the perimeters.
  • Comparison: Once we calculate the perimeters, we will compare them to understand the difference in ribbon lengths needed for the two pillows.

Understanding these key concepts is essential for setting up the problem correctly and choosing the right approach to solve it. Now, let's move on to the calculations!

Calculating the Perimeter of the First Pillow

The first step in solving this problem is to calculate the perimeter of the first pillow. We know that the side length of the first pillow is given by the expression 2x² + 1 inches. As we discussed earlier, the perimeter of a square is four times the length of its side. Therefore, to find the perimeter of the first pillow, we need to multiply the expression 2x² + 1 by 4.

Perimeter Calculation:

  • Side length of the first pillow: 2x² + 1 inches
  • Perimeter of the first pillow: 4 * (2x² + 1)

To simplify this expression, we will use the distributive property, which states that a(b + c) = ab + ac. Applying this property, we get:

  • 4 * (2x² + 1) = 4 * 2x² + 4 * 1
    • = 8x² + 4*

So, the perimeter of the first pillow is 8x² + 4 inches. This expression tells us the total length of ribbon Margot needs for the first pillow in terms of x. It's important to remember that x is a variable, and its value will affect the actual perimeter. However, for now, we have a general expression that represents the perimeter. Now that we've calculated the perimeter of the first pillow, let's move on to the second pillow.

Calculating the Perimeter of the Second Pillow

Now, let's focus on the second pillow. The side length of the second pillow is given as 4x - 7 inches. Similar to the first pillow, we need to find the perimeter by multiplying the side length by 4, since a square has four equal sides.

Perimeter Calculation:

  • Side length of the second pillow: 4x - 7 inches
  • Perimeter of the second pillow: 4 * (4x - 7)

Again, we'll use the distributive property to simplify this expression. This means we multiply 4 by each term inside the parentheses:

  • 4 * (4x - 7) = 4 * 4x - 4 * 7
    • = 16x - 28*

Therefore, the perimeter of the second pillow is 16x - 28 inches. This expression gives us the total ribbon length needed for the second pillow, also in terms of the variable x. We now have expressions for the perimeters of both pillows. The next step is to compare these perimeters, which will help us determine which pillow requires more ribbon and by how much.

Comparing the Perimeters

Now that we have calculated the perimeters of both pillows, it's time to compare them. This will help us understand the difference in the amount of ribbon Margot needs for each pillow. The perimeter of the first pillow is 8x² + 4 inches, and the perimeter of the second pillow is 16x - 28 inches. To compare these, we can look at the expressions and consider how they might differ depending on the value of x.

Comparison Strategy:

  • Identify Key Differences: Notice that the first pillow's perimeter expression includes a squared term (x²), while the second pillow's expression is linear (just x). This means the first pillow's perimeter will grow much faster as x increases.
  • Consider Different Values of x: To get a better sense of the comparison, we could try plugging in a few different values for x and see how the perimeters change. This will give us some specific numerical examples.
  • Find the Difference: To find exactly how much longer one ribbon needs to be compared to the other, we can subtract the perimeter of the second pillow from the perimeter of the first pillow. This will give us an expression for the difference in ribbon length.

Let's start by finding the difference in perimeters. This will give us a clear expression that represents how much more ribbon is needed for one pillow compared to the other.

Finding the Difference in Perimeters

To find the difference in perimeters, we need to subtract the expression for the perimeter of the second pillow from the expression for the perimeter of the first pillow. Remember, the perimeter of the first pillow is 8x² + 4 inches, and the perimeter of the second pillow is 16x - 28 inches.

Difference Calculation:

  • Difference = (Perimeter of first pillow) - (Perimeter of second pillow)
  • Difference = (8x² + 4) - (16x - 28)

To simplify this expression, we need to distribute the negative sign to both terms inside the second parentheses:

  • Difference = 8x² + 4 - 16x + 28

Now, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have the constant terms 4 and 28, which we can add together:

  • Difference = 8x² - 16x + 4 + 28
  • Difference = 8x² - 16x + 32

So, the difference in perimeters is 8x² - 16x + 32 inches. This expression tells us exactly how much longer the ribbon for the first pillow needs to be compared to the ribbon for the second pillow. We now have a clear mathematical representation of the difference, which is a significant step in understanding the problem.

Analyzing the Difference

Now that we have the expression for the difference in perimeters, 8x² - 16x + 32 inches, let's analyze what this means. This expression represents the additional length of ribbon needed for the first pillow compared to the second pillow. To fully understand this, we can explore how this difference changes for different values of x.

Understanding the Expression:

  • Quadratic Nature: The expression is quadratic because it includes an x² term. This means the difference in perimeters will change in a non-linear way as x changes.
  • Positive Coefficient: The coefficient of the x² term is positive (8), which means the parabola opens upwards. This suggests that for very large or very small values of x, the difference will be positive, meaning the first pillow will need more ribbon.
  • Finding the Minimum: To find the minimum difference, we could complete the square or use calculus. However, for the purpose of this problem, we can also consider some specific values of x to get a sense of the range.

Let's think about what values of x make sense in this context. Remember that the side length of the second pillow is 4x - 7 inches. Since a side length cannot be negative, we must have 4x - 7 > 0. Solving this inequality gives us x > 7/4 or x > 1.75. So, we can consider values of x greater than 1.75 to make practical sense of the problem.

Conclusion

In conclusion, we've successfully navigated through a mathematical problem involving pillows and ribbons! We started by understanding the problem, which involved calculating and comparing the perimeters of two square pillows. We then calculated the perimeters of both the first pillow (8x² + 4 inches) and the second pillow (16x - 28 inches) using algebraic expressions and the formula for the perimeter of a square. By finding the difference in perimeters (8x² - 16x + 32 inches), we were able to determine the additional ribbon length needed for the first pillow compared to the second pillow. This problem highlights the practical applications of algebra in everyday scenarios, such as sewing projects or home improvement tasks. By understanding how to use algebraic expressions to represent real-world situations, we can solve a variety of problems and make informed decisions. Remember, mathematics is not just about numbers and equations; it's a powerful tool that helps us understand and interact with the world around us.

For further exploration on mathematical concepts and problem-solving strategies, you can visit Khan Academy. They offer a wide range of resources to enhance your mathematical skills.