Environmental Plume Mapping (Environmental Engineering)

At an industrial manufacturing complex in North Las Vegas, a subterranean transmission line rupture releases an aqueous chemical contaminant onto a flat, horizontal impermeable clay confining layer. Autonomous in-situ geophysical sensors record that the spilled contaminant spreads symmetrically outward in a circular surface plume.

Telemetry monitoring establishes that the surface area $A$ of the contaminant plume expands at a constant rate of:

An emergency environmental remediation team from the Nevada Division of Environmental Protection is deploying a circular containment boom and soil-vapor extraction perimeter to neutralize the hazardous plume before it migrates toward nearby municipal groundwater conduits.

1. Formulate the geometric relationship connecting the surface area $A$ and the radial boundary $r$ of the plume, and calculate the time derivative relation connecting $\frac{dA}{dt}$ and $\frac{dr}{dt}$ using the Chain Rule.
2. Determine the exact instantaneous rate at which the outer radius of the toxic plume is expanding ($\frac{dr}{dt}$) at the precise moment the plume radius reaches $r = 15.0\text{ meters}$.
3. Calculate the required radial expansion speed of the perimeter when the contaminated surface area reaches $A = 400\pi\text{ m}^2$, and explain why remediation engineers must calculate radial velocities rather than relying solely on area rates to position physical containment barriers.