1. **Problem statement:** A building measures 80 m by 50 m, and a paved road of uniform width surrounds it. The area of the paved road alone is 7850 m². We need to find the width of the paved road. 2. **Formula and explanation:** Let the width of the paved road be $x$ meters. The total dimensions including the paved road will be: - Length: $80 + 2x$ - Width: $50 + 2x$ The total area including the paved road is: $$ (80 + 2x)(50 + 2x) $$ The area of the building alone is: $$ 80 \times 50 = 4000 $$ The area of the paved road only is given as 7850 m². So, the total area minus the building area equals the paved area: $$ (80 + 2x)(50 + 2x) - 4000 = 7850 $$ 3. **Set up the equation:** $$ (80 + 2x)(50 + 2x) = 7850 + 4000 = 11850 $$ 4. **Expand the left side:** $$ 80 \times 50 + 80 \times 2x + 2x \times 50 + 2x \times 2x = 11850 $$ $$ 4000 + 160x + 100x + 4x^2 = 11850 $$ 5. **Simplify:** $$ 4x^2 + 260x + 4000 = 11850 $$ 6. **Bring all terms to one side:** $$ 4x^2 + 260x + 4000 - 11850 = 0 $$ $$ 4x^2 + 260x - 7850 = 0 $$ 7. **Divide entire equation by 2 for simplicity:** $$ 2x^2 + 130x - 3925 = 0 $$ 8. **Use quadratic formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=2$, $b=130$, $c=-3925$. Calculate discriminant: $$ \Delta = 130^2 - 4 \times 2 \times (-3925) = 16900 + 31400 = 48300 $$ Calculate square root: $$ \sqrt{48300} \approx 219.77 $$ 9. **Calculate roots:** $$ x = \frac{-130 \pm 219.77}{4} $$ Two solutions: - $$ x_1 = \frac{-130 + 219.77}{4} = \frac{89.77}{4} = 22.44 $$ - $$ x_2 = \frac{-130 - 219.77}{4} = \frac{-349.77}{4} = -87.44 $$ Since width cannot be negative, discard $x_2$. 10. **Final answer:** The width of the paved road is approximately $22.4$ meters (rounded to the nearest tenth).