The problem states:

A rectangular pool measuring $12$ meters by $8$ meters is surrounded by a walkway of width $x$ meters. The question asks for the value of $x$ such that the area of the walkway is equal to the area of the pool.

Step-by-Step Solution:

1. Calculate the Area of the Pool:

   • The pool is $12$ meters long and $8$ meters wide.
   • Area of the pool $= 12 \times 8 = 96$ square meters.

2. Calculate the Dimensions of the Pool + Walkway:

   • The walkway surrounds the pool on all sides, adding $x$ meters to each dimension.
   • So, the total length including the walkway is $12 + 2x$.
   • The total width including the walkway is $8 + 2x$.

3. Calculate the Total Area of Pool + Walkway:

   • The area of the pool and walkway combined is given by the area of a rectangle with dimensions $(12 + 2x) \times (8 + 2x)$.
   • Expanding this: $\text{Total area} = (12 + 2x)(8 + 2x) = 96 + 24x + 16x + 4x^2 = 96 + 40x + 4x^2$

4. Calculate the Area of the Walkway Alone:

   • The area of the walkway is the total area of the pool and walkway minus the area of the pool.
   • Thus, the area of the walkway is: $\text{Area of walkway} = (96 + 40x + 4x^2) - 96 = 40x + 4x^2$

5. Set Up the Equation:

   • We want the area of the walkway to equal the area of the pool: $40x + 4x^2 = 96$

6. Solve the Equation:

   • Rearrange it into standard quadratic form: $4x^2 + 40x - 96 = 0$
   • Divide by 4 to simplify: $x^2 + 10x - 24 = 0$
   • Factor the quadratic equation: $(x + 12)(x - 2) = 0$
   • So, $x = -12$ or $x = 2$. Since a walkway width cannot be negative, $x = 2$ meters.

Conclusion:

The value of $x$ that makes the area of the walkway equal to the area of the pool is 2 meters.

Would you like further details or explanations on any part of the solution?

Related Questions:

1. How would the solution change if the pool were circular?
2. What if the walkway's width varied along each side? How would we approach the problem then?
3. How can you determine the perimeter of the walkway?
4. How would the area change if the pool dimensions were doubled?
5. What would the answer be if the walkway area were to be half the pool's area?

Tip:

When dealing with walkway or border problems, always consider the added dimensions carefully to set up equations correctly.