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A path crosses a rectangular field on a diagonal.Someone travels across the field on the diagonal, instead of walking along the sides, what is the greatest possible percent reduction in total distance traveled?Express your answer to nearest number.

Mar 21st, 2015

Let x, y be the sides of the rectangle and d be its diagonal. They form a right triangle where by θ we denote the angle opposite to the side y. By using trigonometric functions, write x = d cosθ and y = d sinθ.

By walking across on the diagonal the distance reduces by x + y - d = d ( cosθ + sinθ - 1), the percentage of reduction being (x + y - d) / (x + y) =  ( cosθ + sinθ - 1)/ ( cosθ + sinθ) = 1 - 1/( cosθ + sinθ) =

1 -  1/sqrt{2} sin(θ+π/4) <= 1 - 1/sqrt{2} = 1 - 0.7071 = 0.2929 = 29%.

We used here the formula cosθ + sinθ = sqrt{2} sin(θ+π/4) and the fact that sine takes the values between -1 and 1. Note that the maximum reduction is achieved when θ = π/4 (45  de9%grees), and the rectangle turns into a square.

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Mar 21st, 2015

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Mar 21st, 2015
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