Select the correct answer.

Annalise makes rectangular patchwork quilts out of leftover fabric scraps. The first scrap she is using to make a quilt has a width of [tex]2x[/tex] feet and a length of [tex]5x[/tex] feet. When finished, the entire quilt will be 5 feet wider and 3 feet longer than the first fabric scrap used. Which of the following functions will give the area, [tex]f(x)[/tex], of the quilt in square feet?

A. [tex]f(x) = 10x^2 + 31x + 15[/tex]
B. [tex]f(x) = 10x^2 + 16x + 15[/tex]
C. [tex]f(x) = 10x^2 + 25x + 15[/tex]
D. [tex]f(x) = 10x^2 + 15[/tex]



Answer :

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]