1952 AMC 12 第 43 题

Question

将一个圆的直径分成

n

等份，并以每一份为直径作半圆。当

n

变得非常大时，这些半圆的弧长之和趋近于：

The diameter of a circle is divided into

n

equal parts. On each part a semicircle is constructed. As

n

becomes very large, the sum of the lengths of the arcs of the semicircles approaches a length:

原圆周长的一半

Equal to the semi-circumference of the original circle

原圆的直径

Equal to the diameter of the original circle

大于原圆的直径，但小于原圆周长的一半

Greater than the diameter but less than the semi-circumference of the original circle

无穷大

That is infinite

大于原圆周长的一半，但为有限值

Greater than the semi-circumference but finite

答案：A

知识点：圆弧圆周长

难度评级：1360

小提示：

设原圆直径为

d

，则每个小半圆的直径为

\frac{d}{n}

Let the original diameter be

d

, so each small semicircle has diameter

\frac{d}{n}

大提示：

将一个小半圆的弧长乘以份数

n

Multiply the arc length of one small semicircle by the number

n

of parts

解答：

每个小半圆的直径为

\frac{d}{n}

，所以弧长为

\frac{\pi d}{2n}

。全部

n

条弧的长度之和为

n\cdot\frac{\pi d}{2n}=\frac{\pi d}{2}\text{，}

恰好等于原圆周长的一半。这个等式对每个正

n

都成立，而不仅仅在极限情况下成立。

因此，正确答案是 A。

Each small semicircle has diameter

\frac{d}{n},

so its arc length is

\frac{\pi d}{2n}.

The sum of all

n

arc lengths is

n\cdot\frac{\pi d}{2n}=\frac{\pi d}{2},

exactly the semi-circumference of the original circle. This equality holds for every positive

n,

not merely in the limit.

Thus, the correct answer is A.

1952 AMC 12 第 43 题

答案：A

难度评级：1360

小提示：

大提示：

解答：

其他年份的第 43 题