### Question Description

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.

## Final Answer

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.

Answer: 29%.

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