1. **Problem statement:** We have a rectangular pool measuring 30 meters by 6 meters surrounded by a uniform path of width $x$ meters. We want to express the area of the path, $A(x)$, as a function of $x$. 2. **Understanding the problem:** The total area including the pool and the path is a larger rectangle with dimensions $(30 + 2x)$ by $(6 + 2x)$ because the path adds $x$ meters on each side. 3. **Formula for area of a rectangle:** The area is length times width. So, the total area including the path is: $$ (30 + 2x)(6 + 2x) $$ 4. **Area of the pool alone:** The pool area is: $$ 30 \times 6 = 180 $$ 5. **Area of the path alone:** The path area is the total area minus the pool area: $$ A(x) = (30 + 2x)(6 + 2x) - 180 $$ 6. **Expand the product:** $$ (30 + 2x)(6 + 2x) = 30 \times 6 + 30 \times 2x + 2x \times 6 + 2x \times 2x = 180 + 60x + 12x + 4x^2 $$ 7. **Simplify the expression:** $$ 180 + 72x + 4x^2 $$ 8. **Subtract the pool area:** $$ A(x) = (180 + 72x + 4x^2) - 180 = 72x + 4x^2 $$ 9. **Final answer:** $$ A(x) = 4x^2 + 72x $$ This function gives the area of the path in square meters as a function of the path width $x$ meters.