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 ) = e x 85 f(x) = \displaystyle \sqrt[85]{{e}^{x}} . He starts at the point ( 0 , 1 ) (0,1) .

Although 2D-Calvin is a skilled mountain climber, he cannot climb surfaces that have an inclination greater than 8 5 ∘ 85^{\circ} , and finishes his climbing by the point ( x 0 , y 0 ) (x_0 , y_0) .

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


The answer is 1134.

Guilherme Dela Corte by Guilherme Dela Corte , 24, Brazil

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1 solution

The inclination of the mountain is f ′ ( x ) = e x 85 85 f'(x) = \dfrac{\sqrt[85]{e^x}}{85} , and because Calvin cannot climb when the inclination is greater than 8 5 ∘ 85^{\circ} , we have f ′ ( x ) ≤ tan ⁡ 8 5 ∘ f'(x) \leq \tan{85^{\circ}} and thus f ′ ( x 0 ) = tan ⁡ 8 5 ∘ f'(x_0) = \tan{85^{\circ}} .

Noticing 85 f ′ ( x ) = f ( x ) 85 f'(x) = f(x) , we have f ( x 0 ) = 85 tan ⁡ 8 5 ∘ ≈ 971.551 f(x_0) = 85 \tan{85^{\circ}} \approx 971.551 and x 0 ≈ 584.706 x_0 \approx 584.706 .

By the Pythagorean Theorem, we have that D 2 ≈ ( 584.706 ) 2 + ( 971.551 − 1 ) 2 ⇔ D + 1 ≈ 1134. D^2 \approx (584.706)^2 + (971.551 - 1)^2 \Leftrightarrow \boxed{D+1 \approx 1134.}

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