Agricultural Runoff Retention Wetland Partitioning (Environmental Engineering) Solution

Problem Overview

An environmental engineering team optimizes the geometric layout of a constructed treatment wetland basin with active water area $A = 30,000\text{ m}^2$ . Bounded on one long side by a natural ridge, the layout requires three transverse berm segments of width $x$ and one longitudinal berm of length $y$ . We minimize total berm construction length $L = 3x + y$ .

Step-by-Step Mathematical Solution

Step 1: Formulate Governing Equations

The governing process follows the Applied Optimization Modeling Framework: (Defined in Applied Optimization Modeling Framework.)

Objective Function: Minimize total linear berm length: $L(x, y) = 3x + y$ (3 segments of length $x$ : 2 outer sidewalls + 1 interior baffle; 1 segment of length $y$ : outer longitudinal wall).
Hydraulic Surface Area Constraint: $A = x \cdot y = 30,000\text{ m}^2$

Step 2: Variable Reduction (Substitution)

Solve the area constraint for $y$ :

y = \frac{30,000}{x}

Substitute into the berm length objective:

L(x) = 3x + \frac{30,000}{x} = 3x + 30,000x^{-1}

Step 3: Context-Driven Domain

(Governed by Context-Driven Domain in Geometric Optimization.)

Physical dimensions must be positive: $x > 0$ .
Feasible domain is the open interval $x \in (0, \infty)$ .

Step 4: Differentiate and Find Critical Numbers

Differentiate $L(x)$ with respect to $x$ using the Power Rule:

L'(x) = \frac{d}{dx}\left[3x + 30,000x^{-1}\right] = 3 - \frac{30,000}{x^2}

Set

L'(x) = 0

3 - \frac{30,000}{x^2} = 0 \implies 3x^2 = 30,000 \implies x^2 = 10,000

x = \sqrt{10,000} = 100\text{ meters}

(Reject $x = -100\text{ m}$ as non-physical).

Step 5: Verify Global Minimum via Second Derivative Test

Compute the second derivative:

L''(x) = \frac{d}{dx}\left[3 - 30,000x^{-2}\right] = \frac{60,000}{x^3}

Evaluate at

x = 100\text{ m}

L''(100) = \frac{60,000}{(100)^3} = \frac{60,000}{1,000,000} = 0.06 > 0

Since

L''(100) > 0

across the entire domain

(0, \infty)

, the curve is concave up, proving that

x = 100\text{ m}

is the unique absolute minimum berm length.

Step 6: Compute Optimal Dimensions and Total Berm Length

Parallel Length ( $y$ ): $y = \frac{30,000}{100} = 300\text{ meters}$
Total Berm Length Required: $L = 3(100) + 300 = 600\text{ meters}$

Environmental Design Takeaway: The optimal rectangular aspect ratio is $y = 3x$ (the wetland is three times as long as it is wide). This elongated aspect ratio is simultaneously hydraulically ideal for preventing short-circuiting in plug-flow wastewater treatment.

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