Let's analyze Sam's situation step by step:
1. Dimensions of the Flower Patch:
- The flower patch is 12 feet long and 6 feet wide.
2. Area of the Flower Patch:
- The area of the flower patch is calculated by multiplying its length and its width:
[tex]\[
\text{Area of flower patch} = 12 \times 6 = 72 \text{ square feet}.
\][/tex]
3. Function [tex]\(A(x)\)[/tex]:
- Sam created the function [tex]\(A(x)\)[/tex] to represent the total area taken up by the flower patch and the walkway:
[tex]\[
A(x) = 4x^2 + 36x + 72.
\][/tex]
4. Interpreting [tex]\(A(x)\)[/tex]:
- The function [tex]\(A(x)\)[/tex] can be compared to the expansion of the product [tex]\((\text{new length}) \times (\text{new width})\)[/tex]:
- The new length of the flower patch plus the walkway is [tex]\((12 + 2x)\)[/tex].
- The new width of the flower patch plus the walkway is [tex]\((6 + 2x)\)[/tex].
Expanding [tex]\((12 + 2x)(6 + 2x)\)[/tex]:
[tex]\[
(12 + 2x)(6 + 2x) = 12 \times 6 + 12 \times 2x + 2x \times 6 + 2x \times 2x = 72 + 24x + 12x + 4x^2 = 4x^2 + 36x + 72.
\][/tex]
5. Identifying Components in [tex]\(A(x)\)[/tex]:
- In the function [tex]\(A(x) = 4x^2 + 36x + 72\)[/tex]:
- The term [tex]\(72\)[/tex] represents the original area of the flower patch.
- The terms [tex]\(4x^2 + 36x\)[/tex] represent the additional area added by the walkway.
6. Conclusion:
- The combined terms [tex]\(4x^2 + 36x\)[/tex] specifically represent the area added by the walkway.
Therefore, the correct answer is:
C. the total area of the walkway
