The problem in the image describes a rectangular garden measuring 21 feet long and 16 feet wide. A brick walkway of uniform width is constructed around the garden, and the total area of the walkway is 263 square feet. We are tasked with finding the width of the walkway.

Step-by-Step Solution:

1. Area of the Garden: The area of the rectangular garden is: $A_{\text{garden}} = \text{length} \times \text{width} = 21 \times 16 = 336 \, \text{square feet}$

2. Total Area Including the Walkway: The garden is surrounded by a walkway of uniform width $x$. This makes the overall dimensions of the garden plus walkway:

   • Length: $21 + 2x$ (since the walkway adds $x$ feet on both sides of the garden)
   • Width: $16 + 2x$

   So, the total area (garden + walkway) is: $A_{\text{total}} = (21 + 2x)(16 + 2x)$

3. Area of the Walkway: The area of the walkway alone is the difference between the total area and the garden's area: $A_{\text{walkway}} = A_{\text{total}} - A_{\text{garden}} = 263 \, \text{square feet}$

   This gives the equation: $(21 + 2x)(16 + 2x) - 336 = 263$

4. Simplify and Solve the Equation: Expanding the left-hand side: $(21 + 2x)(16 + 2x) = 336 + 42x + 32x + 4x^2 = 336 + 74x + 4x^2$ Substituting this into the equation: $336 + 74x + 4x^2 - 336 = 263$ Simplifying further: $74x + 4x^2 = 263$ Rearranging to get a standard quadratic equation: $4x^2 + 74x - 263 = 0$

5. Solve the Quadratic Equation: We can solve this using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the equation $4x^2 + 74x - 263 = 0$, we have:

   • $a = 4$
   • $b = 74$
   • $c = -263$

   Plugging these values into the quadratic formula: $x = \frac{-74 \pm \sqrt{74^2 - 4(4)(-263)}}{2(4)}$

   I'll calculate the result.The two possible solutions for the width $x$ of the walkway are:

• $x_1 = 3.051$ feet (rounded to the nearest thousandth)
• $x_2 = -21.551$ feet

Since a negative width does not make sense in this context, the correct width of the walkway is:

$x = 3.051 \, \text{feet}$

Let me know if you need any further details or explanations!

Follow-up questions:

1. How would the problem change if the garden's dimensions were different?
2. What happens to the walkway's width if the total walkway area was larger or smaller?
3. Can you generalize a formula to find the width of the walkway for any rectangular garden and walkway area?
4. What other methods, besides the quadratic formula, could be used to solve this problem?
5. How does the addition of the walkway affect the total perimeter of the garden?

Tip:

When solving quadratic equations, always check the context to determine which solution (if more than one) makes sense.