margot is sewing a ribbon on a seam along the perimeter of a square pillow. the side length of the pillow is…

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

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

Answer

Explanation:

Step1: Find perimeters of each pillow

The perimeter of a square is $4s$, where $s$ is the side - length. For the first square pillow with side - length $s_1 = 2x^{2}+1$, its perimeter $P_1=4(2x^{2}+1)=8x^{2}+4$. For the second square pillow with side - length $s_2 = 4x - 7$, its perimeter $P_2=4(4x - 7)=16x-28$.

Step2: Find total perimeter for both pillows

The total length of ribbon needed for both pillows $P = P_1+P_2$. $P=(8x^{2}+4)+(16x - 28)=8x^{2}+16x-24 = 4(2x^{2}+4x - 6)$.

Step3: Substitute $x = 3.5$

Substitute $x = 3.5$ into the expression $4(2x^{2}+4x - 6)$. First, calculate the value inside the parentheses: $2x^{2}+4x - 6=2\times(3.5)^{2}+4\times3.5 - 6$. $2\times(3.5)^{2}=2\times12.25 = 24.5$, $4\times3.5 = 14$. $24.5+14 - 6=32.5$. Then, $4\times32.5 = 130$.

Answer:

D. $4(2x^{2}+4x - 6);130.0$ inches