ee sides of a rectangular paddock are to be fenced, the fourth side being an existing straight er drain. If 1000 m of fencing is available, what dimensions should be used for the paddock to cimise its area?

To maximize the area of the rectangular paddock with three sides fenced and one side (the drain) open, we can follow these steps: 1. **Define Variables:** - Let \( L \) be the length of the paddock parallel to the drain. - Let \( W \) be the width of the paddock perpendicular to the drain. 2. **Perimeter Constraint:** Since only three sides need fencing: \[ L + 2W = 1000 \text{ meters} \] Solving for \( L \): \[ L = 1000 - 2W \] 3. **Area to Maximize:** The area \( A \) of the rectangle is: \[ A = L \times W = (1000 - 2W) \times W = 1000W - 2W^2 \] 4. **Optimization:** To find the maximum area, take the derivative of \( A \) with respect to \( W \) and set it to zero: \[ \frac{dA}{dW} = 1000 - 4W = 0 \] Solving for \( W \): \[ W = 250 \text{ meters} \] Substitute \( W = 250 \) meters back into the perimeter equation to find \( L \): \[ L = 1000 - 2(250) = 500 \text{ meters} \] 5. **Conclusion:** The dimensions that maximize the area of the paddock are: - **Length (\( L \))**: 500 meters - **Width (\( W \))**: 250 meters **Thus, to maximize the area, the paddock should be 500 m long and 250 m wide.**