Environmental Plume Mapping (Environmental Engineering) Solution

Problem Overview

We analyze an expanding circular chemical contaminant plume spreading across an impermeable geological barrier layer with a constant volumetric surface area growth rate of $\frac{dA}{dt} = 12.0\text{ m}^2/\text{hr}$. We calculate the instantaneous radial velocity $\frac{dr}{dt}$ at specific radial snapshots to establish emergency containment boundaries.

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Step-by-Step Mathematical Solution

Step 1: Formulate the Geometric Model & Time Derivative

The governing formula is the Area of a Circle:

(This formula and its time derivative principles are defined in Related Rates.)

Because the contaminant spreads continuously over time, both the area $A(t)$ and the radius $r(t)$ are dynamic variables depending implicitly on time $t$. Applying the time differential operator $\frac{d}{dt}$ to both sides using the Chain Rule:

Isolating the instantaneous radial expansion rate $\frac{dr}{dt}$:

Step 2: Calculate Radial Expansion Speed at $r = 15.0\text{ m}$

Substitute the given telemetry values ($r = 15.0\text{ m}$ and $\frac{dA}{dt} = 12.0\text{ m}^2/\text{hr}$):

Converting to centimeters per hour for field technicians:

Step 3: Evaluate Radial Velocity when Area Reaches $A = 400\pi\text{ m}^2$

1. Determine the snapshot radius $r$:
   $$A = \pi r^2 \implies 400\pi = \pi r^2 \implies r^2 = 400 \implies r = 20.0\text{ m}$$
2. Compute $\frac{dr}{dt}$ at $r = 20.0\text{ m}$:
   $$\frac{dr}{dt} = \frac{12.0}{2\pi (20.0)} = \frac{12.0}{40.0\pi} = \frac{3}{10\pi}\text{ m/hr} \approx 0.0955\text{ m/hr} \approx 9.55\text{ cm/hr}$$

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Engineering Conclusion

In environmental contamination response and remediation engineering, knowing the areal influx rate ($\frac{dA}{dt}$) is necessary for estimating the total mass of leaked contaminant, but it is insufficient for physical containment deployment. Physical containment equipment—such as absorbent surface booms, trench cut-off walls, and sheet-pile barriers—must be positioned at a specific radial distance $r$ from the spill source.

Because $\frac{dr}{dt} = \frac{1}{2\pi r}\frac{dA}{dt}$, the radial expansion velocity is inversely proportional to the radius. When the plume is small ($r = 5\text{ m}$), the boundary advances rapidly ($\approx 38.2\text{ cm/hr}$), posing an immediate threat of overrunning field crews. As the plume expands to $r = 20\text{ m}$, the radial velocity drops to $9.55\text{ cm/hr}$. Calculating $\frac{dr}{dt}$ gives remediation engineers the exact time window available to excavate cutoff trenches safely ahead of the advancing plume boundary.