Question

一个球从 $3$ 米高处落下。第一次反弹时它升到 $2$ 米高。之后每次下落并反弹到前一次反弹高度的 $\frac{2}{3}$ 。第几次反弹时它不会升到 $0.5$ 米高？

A ball is dropped from a height of $3$ meters. On its first bounce it rises to a height of $2$ meters. It keeps falling and bouncing to $\frac{2}{3}$ of the height it reached in the previous bounce. On which bounce will it not rise to a height of $0.5$ meters?

$\ 3$

$\ 4$

$\ 5$

$\ 6$

$\ 7$

答案：C

知识点：等比数列

难度评级：1150

解答：

第一次反弹升到 $2$ 米。之后每次反弹都是前一次反弹高度的 $\dfrac{2}{3}$ 。

第四次反弹升到 $2\left(\dfrac{2}{3}\right)^3=\dfrac{16}{27}$ ，大于 $\dfrac{1}{2}$ 。

第五次反弹升到 $2\left(\dfrac{2}{3}\right)^4=\dfrac{32}{81}$ ，小于 $\dfrac{1}{2}$ 。

所以正确答案是 C。

The first bounce rises to $2$ meters. Each later bounce is $\dfrac{2}{3}$ of the previous bounce.

The fourth bounce rises to $2\left(\dfrac{2}{3}\right)^3=\dfrac{16}{27}$ , which is greater than $\dfrac{1}{2}$ .

The fifth bounce rises to $2\left(\dfrac{2}{3}\right)^4=\dfrac{32}{81}$ , which is less than $\dfrac{1}{2}$ .

Thus, C is the correct answer.

2008 AMC 8 第 12 题

答案：C

难度评级：1150

解答：

其他年份的第 12 题