To solve the problem of finding the area function [tex]\( f(x) \)[/tex] of Annalise's finished quilt, we need to follow these steps:
1. Determine the expressions for the initial dimensions of the first scrap of fabric:
- Width: [tex]\( W_{\text{initial}} = 2x \)[/tex]
- Length: [tex]\( L_{\text{initial}} = 5x \)[/tex]
2. Add the extra dimensions to get the final dimensions of the quilt:
- The quilt is 5 feet wider than the initial width, so the final width is:
[tex]\[
W_{\text{final}} = W_{\text{initial}} + 5 = 2x + 5
\][/tex]
- The quilt is 3 feet longer than the initial length, so the final length is:
[tex]\[
L_{\text{final}} = L_{\text{initial}} + 3 = 5x + 3
\][/tex]
3. Find the area function [tex]\( f(x) \)[/tex] of the finished quilt:
- The area of a rectangle is given by the product of its width and length. Thus, the area [tex]\( A \)[/tex] as a function of [tex]\( x \)[/tex] is:
[tex]\[
f(x) = W_{\text{final}} \times L_{\text{final}} = (2x + 5) \times (5x + 3)
\][/tex]
4. Expand the product to find the area function [tex]\( f(x) \)[/tex]:
- Use the distributive property (FOIL method) to expand the product:
[tex]\[
(2x + 5) \times (5x + 3) = (2x \times 5x) + (2x \times 3) + (5 \times 5x) + (5 \times 3)
\][/tex]
- Multiply the terms:
[tex]\[
(2x \times 5x) = 10x^2
\][/tex]
[tex]\[
(2x \times 3) = 6x
\][/tex]
[tex]\[
(5 \times 5x) = 25x
\][/tex]
[tex]\[
(5 \times 3) = 15
\][/tex]
- Combine like terms:
[tex]\[
10x^2 + 6x + 25x + 15 = 10x^2 + 31x + 15
\][/tex]
Therefore, the function that gives the area [tex]\( f(x) \)[/tex] of the quilt in square feet is:
[tex]\[
\boxed{f(x) = 10x^2 + 31x + 15}
\][/tex]
