Regional Wastewater Effluent Pipeline Routing (Civil & Environmental Engineering)

In civil and environmental engineering infrastructure design, routing utility pipelines across environmentally sensitive terrain requires balancing heavy construction capital costs against environmental preservation. Laying pipelines across protected wetlands requires specialized trenchless horizontal directional drilling (HDD), biological monitoring, and secondary containment liners, making wetland installation significantly more expensive per linear meter than standard open-trench excavation along dry utility rights-of-way.

A Clark County civil engineering design team is designing a high-density polyethylene (HDPE) tertiary-treated wastewater effluent pipeline connecting a newly expanded water resource recovery facility $W$ to an outfall discharge terminal $T$ on a regional river.

• The water reclamation facility $W$ is located on the northern boundary of a protected wetland corridor of uniform width $w = 600\text{ meters}$.
• The outfall discharge terminal $T$ is located on the southern bank of the wetland corridor, $2000\text{ meters}$ downstream from the facility.
• Pipeline construction across the sensitive wetland costs $\$400$ per linear meter due to horizontal directional drilling requirements.
• Once the pipeline crosses the wetland and reaches the southern boundary at point $P$, it follows a dry utility right-of-way easement along the riverbank to terminal $T$ at a cost of $\$240$ per linear meter.

       W (Treatment Facility)
       +
       | \
 600 m |  \  d_wetland = sqrt(600^2 + x^2)  [Cost: $400/m]
       |   \
  -----+----+------------------------------------+ Riverbank Easement
       0    P(x, 0)                             T (Outfall Terminal)
            |<---------- 2000 - x ------------->| [Cost: $240/m]
            |<-------------- 2000 m ----------->|

1. Formulate the total capital construction cost function $C(x)$ as a single-variable function of the downstream crossing distance $x$.
2. State the physical context-driven domain for $x$.
3. Differentiate $C(x)$ and locate all interior critical numbers.
4. Show how the critical number equation embodies the principles of Fastest Path Across Media and Snell's Law.
5. Apply the Closed Interval Method to evaluate the critical point against the domain boundary endpoints ($x = 0$ and $x = 2000$), identifying the optimal route coordinates and the total capital cost savings.