Question

若干个相扣的环，每个厚 $1$ cm，挂在一个钉子上。最上面的环外径为 $20$ cm。其余每个环的外径都比上方的环小 $1$ cm。最下面的环外径为 $3$ cm。从最上面环的顶部到最下面环的底部的距离是多少 cm？

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：1370

解答：

最上面的环跨越 $20$ cm。下面每个环与上方的环重叠 $2$ cm（两倍的 $1$ -cm 厚度），所以它增加的长度为外径减 $2$ 。

下面各环外径为 $19, 18, \ldots, 3$ ，贡献 $17, 16, \ldots, 1$ 。因此总距离为 $\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173 \text{ cm}. \end{gathered}$

因此，正确答案是 B。

The top ring spans $20$ cm. Each ring below overlaps the ring above by $2$ cm (twice the $1$ -cm thickness), so it adds its outside diameter minus $2.$

The lower rings have outside diameters $19, 18, \ldots, 3,$ contributing $17, 16, \ldots, 1.$ Thus the total distance is $\begin{gathered} 20 + (17 + 16 + \cdots + 1) \\ = 20 + \frac{17 \cdot 18}{2} \\ = 20 + 153 \\ = 173 \text{ cm}. \end{gathered}$

Thus, the correct answer is B.

2006 AMC 12A 第 12 题

答案：B

难度评级：1370

解答：

其他年份的第 12 题