Agricultural Runoff Retention Wetland Partitioning (Environmental Engineering)

Constructed treatment wetlands are engineered ecological systems that treat agricultural runoff and municipal stormwater. To achieve mandated nitrate denitrification and heavy metal settling, environmental engineers must guarantee a minimum water surface area for sunlight exposure and wetland macrophyte contact. However, constructing compacted earthen berm dikes to contain the water is the single largest capital cost of wetland installation.

An environmental engineering firm is designing a rectangular surface-flow treatment wetland with a required active water surface area of $30,000\text{ m}^2$. The wetland is positioned alongside an impermeable natural sandstone ridge that forms one long boundary of length $y$, requiring no constructed embankment.

       ================ Natural Sandstone Ridge (y) ================
      |                                  |                          |
      |       Deep Sediment Forebay      |    Macrophyte Marsh      | x
      |                                  |                          |
       ----------------- Constructed Earthen Berm (y) --------------

The remaining boundary and interior layout require engineered earthen berms:

• Two outer boundary dikes of width $x$ (perpendicular to the ridge).
• One outer boundary dike of length $y$ (parallel to the ridge).
• One interior dividing berm of length $x$ separating the high-velocity sediment forebay from the main emergent marsh basin.

1. State the objective function for total constructed berm length $L(x, y)$ and the hydraulic area constraint.
2. Use the Applied Optimization Modeling Framework to reduce the model to a single-variable function $L(x)$.
3. Determine the Context-Driven Domain for the wetland width $x$.
4. Calculate the dimensions ($x$ and $y$) that minimize the total length of constructed earthen berms.
5. Confirm using the Second Derivative Test that the critical point represents a minimum berm investment, and calculate the total required berm length.