The first scenario is the battle of Cajamarca1, fought in present-day Peru between the Inca Emperor Atahuallpa (with as many as 80,000 Indian infantry) against Francisco Pizarro (with 168 Spaniards: 62 cavalry and 106 infantry). Most accounts indicate that the actual battle was waged between an Inca ceremonial guard of about 7000 warriors and the entire Spaniard force. In eight hours, it is estimated that the Spanish killed 7000 Inca and captured the Emperor, without losing a single man. Armaments for the forces were quite different. The Incas were armed with primitive weapons such as clubs and stone hatchets. They wore quilted armor. The Spaniards had steel helmets and body armor, and steel swords and lances. The Indian quilted armor was effective against blunt instruments such as clubs, but offered little protection against piercing by steel swords and lances. The Spaniard armor was quite effective against the Indians primitive weapons.

Consider the Spanish forces to be S and the Inca forces to be I. In this situation, we can model the battle with the following differential equations:

dS/dt = a S(t) I(t)

dI/dt = b S(t) I(t)

Set the time increment to minutes. What is the killing rate for the Incas? Note, this is the factor a above which indicates Inca attrition of Spanish forces. Use the historical data (above) and estimate the value of b for the above differential equations. Give a plain language explanation of b that is, how would you explain the value of b to a non-analyst? Why are we using the linear law, not the square law?