- The width of the garden is \(w\) and the length is \(w + 3\).
- The 2-foot-wide walkway adds 4 feet to both dimensions: \(w + 2 + 2 = w + 4\) and \(w + 3 + 2 + 2 = w + 7\).
- Multiplying these total dimensions gives \((w + 4)(w + 7)\), representing the total area.
- The correct option is \(\boxed{B}\).
Explanation
- Analysis of the Problem
We are given a rectangular garden with width \(w\) and length \(l\). The length is 3 feet longer than the width, meaning \(l = w + 3\). The garden is surrounded by a 2-foot-wide cement walkway. We want to determine what the algebraic expression \((w+4)(w+7)\) represents. The walkway adds 2 feet to each side of the rectangle, effectively adding 4 feet to both the length and the width.
- Determining Total Dimensions
The inner dimensions of the garden are:
- Width = \(w\)
- Length = \(w + 3\)
The walkway adds 2 feet of space on all four sides of the rectangular garden.
- For the width: There is 2 feet of walkway on the left and 2 feet of walkway on the right. Thus, the total width (garden + walkway) is:
\(w + 2 + 2 = w + 4\).
- For the length: There is 2 feet of walkway on the top and 2 feet of walkway on the bottom. Thus, the total length (garden + walkway) is:
\((w + 3) + 2 + 2 = w + 3 + 4 = w + 7\).
- Interpreting the Expression
The total area of a rectangle is the product of its width and its length.
- Total Width = \(w + 4\)
- Total Length = \(w + 7\)
The total area of the garden and the surrounding walkway is:
\(\text{Area} = \text{Total Width} \times \text{Total Length} = (w + 4)(w + 7)\).
Comparing this to the given expression \((w+4)(w+7)\), we can conclude that the expression represents the total area of the garden and the walkway combined.
- Conclusion
Based on the analysis, the expression \((w+4)(w+7)\) corresponds to the total area of the garden and the surrounding walkway. Comparing this to the given options, it matches option B. The final answer is B.
Examples
In real estate or urban planning, calculating total footprints is essential. For instance, when designing a park with a central playground surrounded by a path, knowing the total area (playground plus path) is necessary to estimate the amount of materials needed for ground coverage, like mulch or concrete. By modeling the dimensions with variables, planners can easily adjust the size of the central area and instantly see how it affects the total project footprint.