To find the width of the walkway, we need to determine the difference between the outside dimensions (including the walkway) and the dimensions of the garden itself.

1. Determine the dimensions of the garden:
   • The area of the garden is given as 72 square feet.
   • Let \(l\) be the length and \(w\) be the width of the garden.
   • Since \(l \times w = 72\), we need to find possible dimensions that fit this area.
2. Express the outside dimensions:
   • The outside dimensions (including the walkway) are given as 12 feet by 18 feet.
3. Set up the equations:
   • Let \(x\) be the width of the walkway.
   • The length of the garden plus two times the width of the walkway equals the outside length: \(l + 2x = 18\).
   • The width of the garden plus two times the width of the walkway equals the outside width: \(w + 2x = 12\).
4. Solve for x:
   • We need to find the dimensions \(l\) and \(w\) such that their product is 72 and they fit within the outside dimensions when the walkway is included.
     Assume \(l\) and \(w\) are such that \(l \leq w\):
   • \(l \times w = 72\)
   • \(l + 2x = 18\)
   • \(w + 2x = 12\)
     Solving these equations simultaneously:
   • From \(l + 2x = 18\), we get \(l = 18 - 2x\).
   • From \(w + 2x = 12\), we get \(w = 12 - 2x\).
     Substitute \(l\) and \(w\) into the area equation:
     \[( 18 - 2x) ( 12 - 2x) = 72\]
5. Expand and solve the quadratic equation:
   \[216 - 36x - 24x + 4x^ 2 = 72\]
   \[4x^ 2 - 60x + 216 = 72\]
   \[4x^ 2 - 60x + 144 = 0\]
   Divide the equation by 4:
   \[x^ 2 - 15x + 36 = 0\]
6. Factor the quadratic equation:
   \[( x - 12) ( x - 3) = 0\]
   So, \(x = 12\) or \(x = 3\).
   Since \(x = 12\) would make the garden dimensions negative (which is not possible), the width of the walkway must be:
   \[x = 3\]

Supplemental Knowledge

When dealing with problems involving areas and dimensions of geometric shapes, it's important to understand the relationships between the length, width, and area. For a rectangle, the area is calculated as:
\[\text { Area} = \text { Length} \times \text { Width} \]
In this task, it's necessary to consider both a garden and its adjoining walkways. We may think of their combined area as being represented as a rectangle that encompasses two smaller rectangles (representing garden and path, respectively), adding width through which both dimensions of garden grow together.

Theory in Practice

Imagine having an exquisite painting to frame that depicts Brianna's garden; your frame could represent her walkway around it. To determine how wide of a frame will fit perfectly around this masterpiece and within its outer dimension (ie: wall space), similar calculations would need to take place just like this problem does.

Understanding geometric concepts like area and perimeter is crucial for solving real-life problems such as designing spaces or creating layouts. At UpStudy, we provide resources that help you master these mathematical concepts through interactive exercises and expert guidance. For detailed insights into geometry or other mathematical topics, explore UpStudy’s live tutor question bank or our AI-powered problem-solving services.
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