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 $x = 3.5$?
\bigcirc $2x^2 + 4x - 6$; 22.0 inches
\bigcirc $2x^2 + 4x - 6$; 32.5 inches
\bigcirc $4(2x^2 + 4x - 6)$; 88.0 inches
\bigcirc $4(2x^2 + 4x - 6)$; 130.0 inches

Explanation:

Step1: Find perimeter of one square

The perimeter of a square is \( 4 \times \text{side length} \). For one pillow, side is \( 2x^2 + 1 \), for the other, \( 4x - 7 \). Total side length for both pillows (since ribbon is along perimeter, we sum the perimeters? Wait, no—wait, the problem says "length of ribbon for both pillows". Wait, no, maybe I misread. Wait, the square pillow: perimeter of one square is \( 4(2x^2 + 1) \), and the similar pillow (also a square, since similar) has perimeter \( 4(4x - 7) \). Wait, no, wait the problem says "the length of ribbon that she needs for both pillows"—so total ribbon is perimeter of first + perimeter of second.

So first perimeter: \( 4(2x^2 + 1) = 8x^2 + 4 \)

Second perimeter: \( 4(4x - 7) = 16x - 28 \)

Total ribbon: \( (8x^2 + 4) + (16x - 28) = 8x^2 + 16x - 24 \). Wait, but let's factor that: \( 8x^2 + 16x - 24 = 8(x^2 + 2x - 3) \)? Wait, no, wait the options have \( 4(2x^2 + 4x - 6) \). Let's check: \( 4(2x^2 + 4x - 6) = 8x^2 + 16x - 24 \), which matches. So that's the expression for total ribbon (sum of perimeters of both squares).

Step2: Substitute x = 3.5

Now, substitute \( x = 3.5 \) into \( 4(2x^2 + 4x - 6) \).

First, calculate inside the parentheses: \( 2x^2 + 4x - 6 \)

\( x = 3.5 \), so \( x^2 = (3.5)^2 = 12.25 \)

\( 2x^2 = 2 \times 12.25 = 24.5 \)

\( 4x = 4 \times 3.5 = 14 \)

So \( 2x^2 + 4x - 6 = 24.5 + 14 - 6 = 32.5 \)

Then multiply by 4: \( 4 \times 32.5 = 130.0 \) inches.

Wait, let's check the options. The last option is \( 4(2x^2 + 4x - 6) \); 130.0 inches. Let's verify the calculation again.

Wait, \( x = 3.5 \):

\( 2x^2 = 2(3.5)^2 = 212.25 = 24.5 \)

\( 4x = 4*3.5 = 14 \)

So \( 2x^2 + 4x - 6 = 24.5 + 14 - 6 = 32.5 \)

Then \( 4*32.5 = 130 \). Yes. So the expression is \( 4(2x^2 + 4x - 6) \) and when \( x=3.5 \), it's 130.0 inches.

Answer:

D. \( 4(2x^2 + 4x - 6) \); 130.0 inches (assuming the last option is D, as per the options: the last option is "4(2x²+4x−6;) 130.0 inches"—probably a typo, but the expression and value match)