When are mathematical models most useful? 

A)  when controlled experiments are impossible 
B)  when controlled experiments are very easy to perform 
C)  when responses to causing events show no variability 
D)  when there are a few interacting variables

Mathematical models are most useful when controlled experiments are impossible, as they allow researchers to simulate and predict outcomes in complex systems without the need for costly, time-consuming and often unrealistic experiments. Mathematical models are especially useful in disciplines where experimentation is expensive or impractical, such as in astronomy, meteorology, economics, or ecology. Furthermore, mathematical models can help identify the key factors or variables that influence a phenomenon, and their simulation can help researchers validate their theories and understand the underlying mechanisms.