“Railroad Track Problem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Railroad Track Problem

释义

Railroad Track Problem

Given a straight segment of track of length , add a small segment so that the track bows into a circular Arc. Find the maximum displacement of the bowed track. The Pythagorean Theorem gives

(1)

But

is simply

, so

(2)

Solving (1) and (2) for

gives

(3)

Expressing the length of the Arc in terms of the central angle,


			(4)

But

is given by

(5)

so plugging

in gives

(6)

(7)

For

(8)

Therefore,


			(9)

Keeping only terms to order

(10)

(11)

(12)

and

(13)

If we take

and

1 foot, then

feet.

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