To determine the function that gives the area of the quilt in square feet, we can follow these steps:
1. Identify dimensions of the first fabric scrap:
- The width is [tex]\(2x\)[/tex] feet.
- The length is [tex]\(5x\)[/tex] feet.
2. Determine dimensions of the finished quilt:
- The width of the quilt will be [tex]\(2x + 5\)[/tex] feet (since it's 5 feet wider).
- The length of the quilt will be [tex]\(5x + 3\)[/tex] feet (since it's 3 feet longer).
3. Write the area function, [tex]\(f(x)\)[/tex]:
- The area of the quilt is given by multiplying its width and length:
[tex]\[
f(x) = (2x + 5)(5x + 3)
\][/tex]
4. Expand the expression:
[tex]\[
(2x + 5)(5x + 3)
\][/tex]
To expand this, we use the distributive property (also known as the FOIL method for binomials):
[tex]\[
= 2x \cdot 5x + 2x \cdot 3 + 5 \cdot 5x + 5 \cdot 3
\][/tex]
[tex]\[
= 10x^2 + 6x + 25x + 15
\][/tex]
5. Combine like terms:
[tex]\[
= 10x^2 + (6x + 25x) + 15
\][/tex]
[tex]\[
= 10x^2 + 31x + 15
\][/tex]
Therefore, the function that gives the area of the quilt in square feet is:
[tex]\[
f(x) = 10x^2 + 31x + 15
\][/tex]
6. Correct answer:
Among the given choices, the correct function is:
A. [tex]\( f(x) = 10x^2 + 31x + 15\)[/tex]
So the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
