1. **State the problem:** A farmer has 200 m of fencing and wants to fence a rectangular field along a straight river. The river side does not need fencing. We need to find the dimensions that maximize the area. 2. **Define variables:** Let $x$ be the length of the side parallel to the river (no fence needed here), and $y$ be the length of the two sides perpendicular to the river. 3. **Write the fencing constraint:** Since the river side needs no fence, fencing is only needed for two $y$ sides and one $x$ side: $$2y + x = 200$$ 4. **Express $x$ in terms of $y$:** $$x = 200 - 2y$$ 5. **Write the area formula:** $$A = x \times y$$ Substitute $x$: $$A = (200 - 2y) y = 200y - 2y^2$$ 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 given by $$y = -\frac{b}{2a} = -\frac{200}{2 \times (-2)} = \frac{200}{4} = 50$$ 8. **Find $x$ at $y=50$:** $$x = 200 - 2 \times 50 = 200 - 100 = 100$$ 9. **Conclusion:** The dimensions that maximize the area are $x = 100$ m (along the river) and $y = 50$ m (perpendicular to the river). 10. **Maximum area:** $$A = 100 \times 50 = 5000$$ So, the maximum area is 5000 square meters with dimensions 100 m by 50 m.