Monitoring Water Quality Across a Filtration System

You are a technician at a municipal water treatment facility responsible for monitoring contaminant concentration levels across a large rectangular filtration bed. The concentration [latex]C[/latex] of a particular compound (measured in parts per million) varies across the bed and is modeled as a function [latex]C(x, y)[/latex], where [latex]x[/latex] and [latex]y[/latex] represent horizontal distances in meters from the southwest corner. Sensors are placed at specific grid locations, but you often need to estimate concentration values at points between sensors. By finding tangent planes to the concentration surface and using linear approximations, you can make reliable estimates without installing additional equipment. Understanding when these approximations are valid—and how to quantify potential changes—is critical for maintaining water quality standards.