Decay and half-life

Build a precise understanding of exponential decay and half-life in precalculus. These problems connect remaining quantity, elapsed time, decay constants and equivalent model forms. The emphasis is on interpreting what a formula means before using logarithms to solve it.

You will calibrate decay from measurements, distinguish the fraction remaining from the fraction lost, reconstruct an earlier amount and compare decay parameters. Error-analysis questions address common mistakes such as subtracting a fixed amount each half-life, assigning the wrong sign to a rate constant or assuming that a mixture of decaying components follows one exponential law.

The examples use synthetic quantities and idealized assumptions; no medical dosing or historical dating claims are required. Prerequisites include exponential equations, logarithm laws, ratios and units. Each explanation shows the defining ratio or equation, checks the result and explains a tempting wrong interpretation. Practice develops from repeated halving to inverse calculations and model criticism. Use scratch work before choosing an answer, and read multiselect statements individually. The results can reveal whether further practice is needed in logarithms, rate conventions or the distinction between a mathematical model and its application.