Rational models, rates and variation

Rational models, rates and variation

Build and interpret rational models for average cost, concentration, travel and work. This precalculus practice connects algebraic formulas with the quantities being divided, the units of the output and the restrictions imposed by the situation. All numerical scenarios are explicitly idealized rather than empirical claims.

Questions include recovering fixed and variable costs from average-cost data, tracking both solute amount and liquid volume, interpreting a finite tank capacity, combining work rates and comparing travel over equal distances. Variation and saturation tasks extend these models to scaling and threshold questions.

Prerequisites are linear and rational equations, proportional reasoning, rational graphs and physical-domain restrictions. Single-select and multiselect items progress from model construction to calibration, combined rates and interpretation. Detailed explanations distinguish an attainable endpoint from an asymptote that is merely approached.

Before calculating, identify the numerator and denominator and attach units to each. Decide whether the input is continuous or discrete, positive or nonnegative, and whether a capacity restricts the model. Then check that the answer fits those conditions. The explanations show why averaging rates directly can fail, why adding liquid can change both parts of a concentration ratio and why a long-run level should not automatically be described as a finite maximum or minimum.