Let's solve this problem step-by-step.

1. Calculate the area of the pool:
   The pool has dimensions $12$ meters by $8$ meters, so its area $A_{\text{pool}}$ is: $A_{\text{pool}} = 12 \times 8 = 96 \text{ square meters}$

2. Express the area of the larger rectangle (pool + walkway):
   Since the walkway surrounds the pool on all sides, the length and width of the larger rectangle (pool + walkway) are increased by $2x$ meters (where $x$ is the width of the walkway) on each dimension:

   • New length: $12 + 2x$
   • New width: $8 + 2x$

   So, the area of the larger rectangle $A_{\text{total}}$ is: $A_{\text{total}} = (12 + 2x)(8 + 2x)$

3. Set up the equation for the walkway area:
   The area of the walkway alone is the area of the larger rectangle minus the area of the pool. Since we want the walkway area to equal the pool area, we set up the equation: $A_{\text{total}} - A_{\text{pool}} = A_{\text{pool}}$ Substituting the values: $(12 + 2x)(8 + 2x) - 96 = 96$

4. Expand and simplify the equation:
   Expanding $(12 + 2x)(8 + 2x)$: $A_{\text{total}} = 96 + 24x + 16x + 4x^2 = 96 + 40x + 4x^2$ Now substitute back: $96 + 40x + 4x^2 - 96 = 96$ Simplifying: $4x^2 + 40x = 96$ Divide by 4: $x^2 + 10x = 24$ Rearrange into standard quadratic form: $x^2 + 10x - 24 = 0$

5. Solve the quadratic equation:
   We can solve $x^2 + 10x - 24 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 10$, and $c = -24$.

   Plugging in these values: $x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}$ $x = \frac{-10 \pm \sqrt{100 + 96}}{2}$ $x = \frac{-10 \pm \sqrt{196}}{2}$ $x = \frac{-10 \pm 14}{2}$

   This gives two solutions: $x = \frac{4}{2} = 2 \quad \text{and} \quad x = \frac{-24}{2} = -12$

   Since $x$ represents a width, we discard the negative solution. Thus, $x = 2$.

Answer:

The width $x$ of the walkway should be 2 meters for the walkway area to equal the area of the pool.

Would you like more details on any of the steps?

Here are some related questions you might find interesting:

1. What if the pool was circular instead of rectangular?
2. How would the solution change if the pool dimensions were different?
3. What would happen if the walkway had varying widths?
4. Can we generalize this formula for any rectangular pool dimensions?
5. What if we needed the walkway area to be twice the pool area?

Tip: When solving similar problems, drawing a diagram can be helpful for visualizing changes in dimensions.