The Mountain Song

Calculus Level pending

In a 2D Cartesian plane world, our friend Calvin wants to climb a mountain that is similar to the function \(f(x) = \displaystyle \sqrt[85]{{e}^{x}}\). He starts at the point \((0,1)\).

Although 2D-Calvin is a skilled mountain climber, he cannot climb surfaces that have an inclination greater than \(85^{\circ}\), and finishes his climbing by the point \((x_0 , y_0)\).

Catching a break so he can go back down, Calvin evaluates, to the nearest integer, his distance \(D\) in meters to the starting point. Find \(D+1\).

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