In environmental and hydrological engineering, subterranean chemical spills threaten regional drinking water aquifers. When an industrial solvent leaks through porous underground gravel and fractured limestone, environmental remediation engineers construct 3D spatial dispersion models to estimate pollutant concentrations, design containment grouting walls, and drill monitoring recovery wells.
An environmental engineering firm in Nevada models the chemical pollutant concentration (in parts per million, ) migrating from a buried ruptured storage pipe using the 3D rational function:
where represent 3D spatial coordinates (measured in meters) relative to an injection test wellhead .
A deep monitoring sensor is installed at the coordinate position .
- State the algebraic domain condition for and write the mathematical domain in set-builder notation. Describe the 3D surface that is physically and mathematically excluded from the domain.
- Determine the mathematical range of the concentration function over its valid domain, analyzing the behavior both inside and outside the excluded boundary surface.
- Evaluate the contaminant concentration at the sensor location . Explain what this value means for environmental safety zones and groundwater monitoring.