1. **State the problem:** We want to find the dimensions of a rectangular field enclosed by 500 meters of fencing, where one side is a building (so no fence needed on that side), that maximize the enclosed area. 2. **Define variables:** Let $x$ be the length of the side parallel to the building (no fence needed), and $y$ be the length of the other two sides perpendicular to the building. 3. **Write the constraint:** Since one side is the building, fencing is needed for the other three sides: $$x + 2y = 500$$ 4. **Express $x$ in terms of $y$:** $$x = 500 - 2y$$ 5. **Write the area function:** $$A = x \times y = (500 - 2y) y = 500y - 2y^2$$ 6. **Maximize the area:** This is a quadratic function in $y$ with a negative coefficient for $y^2$, so it opens downward and has a maximum at the vertex. 7. **Find the vertex:** The vertex $y$-value is at $$y = -\frac{b}{2a} = -\frac{500}{2 \times (-2)} = \frac{500}{4} = 125$$ 8. **Find $x$ corresponding to $y=125$:** $$x = 500 - 2(125) = 500 - 250 = 250$$ 9. **Conclusion:** The dimensions that maximize the area are $x = 250$ meters (side along the building) and $y = 125$ meters (the other two sides). 10. **Maximum area:** $$A = 250 \times 125 = 31250$$ square meters.