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2017 AMC 12A Problem 4

Problem 4 of 25EasierGeometryArithmetic

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia’s trip was, compared to Jerry’s trip?

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Solution

If the square has side x,x, Jerry walks x+x=2x,x+x=2x, while Silvia walks the diagonal x2+x2=x2.\sqrt{x^2+x^2}=x\sqrt2. The fraction by which Silvia’s trip is shorter is 2x−x22x=1−22≈1−0.707=0.293. \begin{aligned} &\dfrac{2x-x\sqrt2}{2x}=1-\dfrac{\sqrt2}{2} \\ &\approx 1-0.707=0.293. \end{aligned} This is closest to 30%.30\%. Thus, the correct answer is A.
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Tagged: Pythagorean Theorem · percentage

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