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:

Step1: Analyze the problem

We have two square pillows. The side length of the first pillow is \(2x^2 + 1\) inches, and the side length of the second (similar) pillow is \(4x - 7\) inches. We need to find the total length of ribbon used for both pillows, which is the sum of their perimeters. The perimeter of a square is \(4\times\) side length.

Step2: Calculate the perimeter of each pillow

• Perimeter of the first pillow: \(P_1=4\times(2x^2 + 1)=8x^2+4\)
• Perimeter of the second pillow: \(P_2 = 4\times(4x - 7)=16x-28\)

Wait, maybe I misread. Wait, the problem says "the side length of the pillow is \(2x^2 + 1\) inches. She plans to make a similar pillow, whose side length is \(4x - 7\) inches. What expression can be used for the length of ribbon that she needs for both pillows..."

Wait, maybe the first pillow's ribbon is along the seam (perimeter) and the second is also a square, so total ribbon is perimeter of first + perimeter of second.

Perimeter of square is \(4s\), where \(s\) is side length.

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

Second perimeter: \(4(4x - 7)\)

Total ribbon: \(4(2x^2 + 1)+4(4x - 7)=4(2x^2 + 1 + 4x - 7)=4(2x^2+4x - 6)\)

Ah, so the expression is \(4(2x^2 + 4x - 6)\)

Now, when \(x = 3.5\), let's compute:

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

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

\(2x^2=2\times12.25 = 24.5\)

\(4x = 4\times3.5 = 14\)

So \(24.5 + 14 - 6=24.5 + 8=32.5\)

Then total ribbon: \(4\times32.5 = 130.0\) inches? Wait, no, wait:

Wait, \(4(2x^2 + 4x - 6)\) when \(x = 3.5\):

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

\(2(3.5)^2 + 4(3.5) - 6\)

\(2*12.25 + 14 - 6\)

\(24.5 + 14 - 6 = 32.5\)

Then multiply by 4: \(4*32.5 = 130.0\) inches. Wait, but let's check the options.

The options are:

1. \(2x^2 + 4x - 6\); 22.0 inches

1. \(2x^2 + 4x - 6\); 32.5 inches

1. \(4(2x^2 + 4x - 6)\); 88.0 inches

1. \(4(2x^2 + 4x - 6)\); 130.0 inches

Wait, let's recalculate \(2x^2 + 4x - 6\) at \(x = 3.5\):

\(2(3.5)^2 + 4(3.5) - 6\)

\(2*12.25 = 24.5\)

\(4*3.5 = 14\)

\(24.5 + 14 = 38.5\); \(38.5 - 6 = 32.5\). So that's the expression inside, but the total ribbon is 4 times that, so \(4*32.5 = 130\). So the expression is \(4(2x^2 + 4x - 6)\) and the length is 130.0 inches. So the correct option is the fourth one: \(4(2x^2 + 4x - 6)\); 130.0 inches

Answer:

1. \(4(2x^2 + 4x - 6)\); 130.0 inches