Question

若干个相连的圆环挂在一个钉子上，每个圆环厚 $1$ 厘米。最上面的圆环外径为 $20$ 厘米。其余每个圆环的外径都比它上方的圆环少 $1$ 厘米。最下面的圆环外径为 $3$ 厘米。从最上面圆环的顶部到最下面圆环的底部的距离是多少厘米？

A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the other rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?

$171$

$173$

$182$

$188$

$210$

答案：B

知识点：等差数列求和

难度评级：1330

解答：

最上面的圆环贡献完整外径 $20$ 厘米。因为圆环厚 $1$ 厘米，每个圆环会比上方圆环的顶部低 $2$ 厘米，所以每个较低圆环增加的是它的外径减去 $2$ 。

外径依次为 $20, 19, \ldots, 3$ ，所以下方各环增加的距离为 $17, 16, \ldots, 1$ 。总距离为 $\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173. \end{gathered}$

所以正确答案是 B。

The top ring contributes its full outside diameter, $20$ cm. Because the rings are $1$ cm thick, each ring hangs $2$ cm below the top of the ring above it, so each lower ring adds its outside diameter minus $2.$

The outside diameters run $20, 19, \ldots, 3,$ so the added distances are $17, 16, \ldots, 1.$ The total is $\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173. \end{gathered}$

Thus, the correct answer is B.

2006 AMC 10A 第 14 题

答案：B

难度评级：1330

解答：

其他年份的第 14 题