Select the correct answer. Sam is installing a walkway around a rectangular flower patch in his garden. The flower patch is 12 feet long and 6 feet wide. The width of the walkway is \( x \) feet. Sam created function \( A \) to represent the total area taken up by the flower patch and walkway by multiplying the functions modeling the new total length and width. A(x) \( =4 x^{2}+36 x+72 \) What does \( 4 x^{2}+36 x \) represent in this function? A. the total area of the flower patch B. the total area of the walkway C. the area of the walkway along the length of the flower patch D. the area of the walkway along the width of the flower patch Next

The expression \( 4x^{2}+36x \) represents the area contribution of the walkway, as it accounts for how the walkway extends the length and width of the flower patch. Specifically, when we expand the area function \( A(x) \), this part indicates the new area created by including the walkway around the existing dimensions of the flower patch. Thus, the correct answer is B: the total area of the walkway. Now, let’s dive a bit deeper into the fun stuff! Did you know that the mathematical concept behind these types of area problems often leads to quadratic functions? Quadratic functions are not only essential in geometry but also appear in various fields such as physics and economics. They help model scenarios like projectile motion and profit maximization! And if you’re intrigued by the geometry of gardens, you might enjoy exploring concepts like the Golden Ratio or Fibonacci Sequence in nature! They not only elevate garden designs but can also inspire art and architecture, as they create visually pleasing proportions. Happy exploring!