Question

margot is sewing a ribbon on a seam along the perimeter of a square pillow. the side length of the pillow is \\(2x^2+1\\) inches. she plans to make a similar pillow, including the ribbon, whose side length is \\(4x-7\\) inches. what expression can be used for the length of ribbon that she needs for both pillows, and what is the length if \\(x = 3.5\\)?

\\(2x^2+4x-6\\); 22.0 inches
\\(2x^2+4x-6\\); 32.5 inches
\\(4(2x^2+4x-6)\\); 88.0 inches
\\(4(2x^2+4x-6)\\); 130.0 inches

Explanation:

Express the perimeter of each square pillow

The perimeter of a square is four times its side length.
For the first pillow with side length \(2x^2 + 1\):

$$P_1 = 4(2x^2 + 1)$$

For the second pillow with side length \(4x - 7\):

$$P_2 = 4(4x - 7)$$

Combine the perimeters for both pillows

Using the Polynomial Addition knowledge point, we sum the two perimeters to find the total ribbon length:

$$
LATEXBLOCK0
$$

Evaluate the expression at \(x = 3.5\)

Substitute \(x = 3.5\) into the simplified total perimeter expression:

$$
LATEXBLOCK1
$$

Answer:

• (A) \(2x^2+4x-6\); \(22.0\) inches
• (B) \(2x^2+4x-6\); \(32.5\) inches
• (C) \(4(2x^2+4x-6)\); \(88.0\) inches
• (D) \(4(2x^2+4x-6)\); \(130.0\) inches (Correct answer)