The Correct Answer and Explanation is:
The correct answer is A. (18-2x)(12-2x)=72.
This problem requires setting up an equation to find the width, x, of a uniform walkway built around a rectangular garden. The key is to express the dimensions of the garden in terms of the overall dimensions and the walkway’s width.
First, let’s identify the information provided in the diagram and text. The problem states that the garden itself has an area of 72 square feet. The diagram shows that the larger rectangle, which includes both the garden and the surrounding walkway, has outer dimensions of 18 feet by 12 feet. The unknown width of the walkway is represented by the variable x.
To find the area of the inner garden, we use the formula Area = Length × Width. We are given that the area is 72. We need to find expressions for the garden’s length and width.
The total length of the large rectangle is 18 feet. This length includes the garden’s length plus the width of the walkway on both the left and right sides. To find the length of just the garden, we must subtract the width of the walkway, x, from each side. Therefore, the garden’s length is 18 – x – x, which simplifies to 18 – 2x.
Similarly, the total width of the large rectangle is 12 feet. This includes the garden’s width plus the walkway at the top and bottom. To find the width of the garden, we subtract x from the top and x from the bottom. This gives us a garden width of 12 – x – x, or 12 – 2x.
Now we can set up the final equation using the garden’s area:
Area of Garden = (Length of Garden) × (Width of Garden)
72 = (18 – 2x)(12 – 2x)
This equation correctly represents the situation described and matches option A.
