To find the total length of the ribbon needed for both pillows, we should start by calculating the perimeter of each square pillow.
1. The side length of the first pillow is [tex]\(2x^2 + 1\)[/tex].
- The perimeter of a square is [tex]\(4\)[/tex] times its side length.
- Therefore, the perimeter of the first pillow is:
[tex]\[
4 \times (2x^2 + 1) = 8x^2 + 4
\][/tex]
2. The side length of the second pillow is [tex]\(4x - 7\)[/tex].
- Similarly, the perimeter of the second pillow is:
[tex]\[
4 \times (4x - 7) = 16x - 28
\][/tex]
3. To find the total length of ribbon needed for both pillows, we add the perimeters of the two pillows:
[tex]\[
(8x^2 + 4) + (16x - 28) = 8x^2 + 16x - 24
\][/tex]
Therefore, the correct expression for the total length of ribbon is:
[tex]\[
8x^2 + 16x - 24
\][/tex]
Now, let's find the total length of the ribbon when [tex]\(x = 3.5\)[/tex]:
1. Substitute [tex]\(x = 3.5\)[/tex] in the expression [tex]\(8x^2 + 16x - 24\)[/tex]:
[tex]\[
8(3.5)^2 + 16(3.5) - 24
\][/tex]
2. Calculate each term:
[tex]\[
8(3.5)^2 = 8 \times 12.25 = 98
\][/tex]
[tex]\[
16(3.5) = 56
\][/tex]
[tex]\[
98 + 56 - 24 = 130
\][/tex]
Therefore, the total length of the ribbon needed when [tex]\(x = 3.5\)[/tex] is [tex]\(130.0\)[/tex] inches.
The correct answer is:
[tex]\[
8x^2 + 16x - 24; 130.0 \text{ inches}
\][/tex]
Thus, the correct option from the given choices is:
[tex]\[
4(2x^2 + 4x - 6); 130.0 \text{ inches}
\][/tex]
