2008 AMC 8 第 12 题

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

小提示：

每次乘以

\frac{2}{3}

，列出反弹高度

List the bounce heights by multiplying by

\frac{2}{3}

each time.

大提示：

将第四次和第五次反弹高度与

\frac{1}{2}

比较

Compare the fourth and fifth bounce heights with

\frac{1}{2}

.

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

小提示：

大提示：

解答：

其他年份的第 12 题