1. **Problem Statement:** A farmer wants to fence a rectangular area with one side along a stream, so only three sides need fencing. The total fence length is 80 m. We need to find the dimensions (length and width) that maximize the enclosed area. 2. **Define variables:** Let $X$ be the length perpendicular to the stream. 3. **Width expression:** Since one side is along the stream, the fence covers two lengths $X$ and one width $W$. Total fence length is: $$W + 2X = 80 \implies W = 80 - 2X$$ 4. **Area formula:** The area $A$ is length times width: $$A = X \times W = X(80 - 2X) = 80X - 2X^2$$ 5. **Maximize area:** To find the maximum area, take the derivative of $A$ with respect to $X$ and set it to zero: $$A'(X) = 80 - 4X$$ Set derivative to zero: $$80 - 4X = 0 \implies 4X = 80 \implies X = 20$$ 6. **Calculate width:** Substitute $X=20$ into width formula: $$W = 80 - 2(20) = 80 - 40 = 40$$ 7. **Calculate maximum area:** $$A = 20 \times 40 = 800$$ **Answer:** The maximum area is 800 square meters with length $20$ m and width $40$ m.