## BONUS Problem From Heck # E

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A Colombian drug lord plans to bribe some police officers so that they will let his cocaine pass the boundary freely. He knows that the more officers he bribes, the more cocaine he can transfer. Let the fraction of bought officers be xϵ[0,1] and let f(x)ϵ[0,1] be the fraction of the total cocaine harvest not intercepted.

Naturally the function f is increasing and f(1) = 1. Indeed, if the whole police is bought than nothing prevents the cocaine from crossing the border.

Suppose that the cocaine is C£ worth and the whole police costs P£. Calculate the maximal income and find out how much police the drug lord should bribe if

\[\large f(x)=\frac{1}{2}+\frac{2}{\pi }\arctan x\]

Why is π in this formula? Well the cocaine grew on a round field.

Naturally the function f is increasing and f(1) = 1. Indeed, if the whole police is bought than nothing prevents the cocaine from crossing the border.

Suppose that the cocaine is C£ worth and the whole police costs P£. Calculate the maximal income and find out how much police the drug lord should bribe if

\[\large f(x)=\frac{1}{2}+\frac{2}{\pi }\arctan x\]

Why is π in this formula? Well the cocaine grew on a round field.

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