We apply Lanchesters model to the Battle of Trafalgar(1805), when a fleet of 40 British

Chapter 11, Problem 52

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We apply Lanchesters model to the Battle of Trafalgar(1805), when a fleet of 40 British ships expected to facea combined French and Spanish fleet of 46 ships. Supposethat there were x British ships and y opposing shipsat time t. We assume that all the ships are identical sothat constants in the differential equations in Lanchestersmodel are equal:dxdt = aydydt = ax.(a) Write a differential equation involving dy/dx, andsolve it using the initial sizes of the two fleets.(b) If the battle were fought until all the British shipswere put out of action, how many French/Spanishships does this model predict would be left at theend of the battle?Admiral Nelson, who was in command of the Britishfleet, did not in fact send his 40 ships against the 46French and Spanish ships. Instead he split the battleinto two parts, sending 32 of his ships against 23 of theFrench/Spanish ships and his other 8 ships against theirother 23.(c) Analyze each of these two sub-battles using Lanchestersmodel. Find the solution trajectory for eachsub-battle. Which side is predicted to win each one?How many ships from each fleet are expected to beleft at the end?(d) Suppose, as in fact happened, that the remainingships from each sub-battle then fought each other.Which side is predicted to win, and with how manyships remaining?

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