# The More, The Mightier

Calculus Level 4

Lanchester's Square Law can be used to roughly describe the way in which two opposing military forces change over time during battle. Suppose the number of troops in "Force A" is $$A$$, and the number of troops in "Force B" is $$B$$.

The rates of change in troop strength (numbers of troops) over time are given by: $$\dfrac{dA}{dt} = -B$$ and $$\dfrac{dB}{dt} = -A$$.

Suppose that $$A=2$$ and $$B=1$$ at time $$(t=0)$$. What is the value of $$A$$ at the moment in time at which $$A=100B$$?

Details and Assumptions:

• Assume that $$A$$ and $$B$$ can vary continuously, and that they are multiples of some standard measure.

• Evidently, the larger force has a distinct advantage if all else is equal.

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