Flying high

An airplane, when in flight, is subjected to drag that is proportional to its speed \(v\). But air flowing over the wings also pushes it downward. Given this consideration, we can model the total air resistance force on a plane as \[F_f = av^2 + bv^{-2}\] for some constants \(a\) and \(b\) which depend on the airplane.

Consider a plane that is in a steady flight at which the engine must provide an equal and opposite force to the resistance force. The speed at which the airplane flies the longest distance given a specific amount of fuel can be expressed as \[v = \left(K \times \frac{b}{a}\right)^\frac{1}{M} \] where \(K\) and \(M\) are positive integers. What is the value of \(K + M \) ?

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