AMC8 2008

AMC8 2008 · Q12

AMC8 2008 · Q12. It mainly tests Exponents & radicals, Fractions.

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?

一个球从3米高处落下。第一次弹起时升到2米高。它不断落下和弹起，每次弹起高度是前一次的$\frac{2}{3}$。在第几次弹起时它不会升到0.5米高？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The table gives the height of each bounce. \[ \begin{array}{|c|c|c|c|c|c|} \hline \text{Bounce} & 1 & 2 & 3 & 4 & 5\\ \hline \text{Height in Meters} & 2 & \frac{2}{3}\cdot 2=\frac{4}{3} & \frac{2}{3}\cdot \frac{4}{3}=\frac{8}{9} & \frac{2}{3}\cdot \frac{8}{9}=\frac{16}{27} & \frac{2}{3}\cdot \frac{16}{27}=\frac{32}{81}\\ \hline \end{array} \] Because $\frac{16}{27}>\frac{16}{32}=\frac{1}{2}$ and $\frac{32}{81}<\frac{32}{64}=\frac{1}{2}$, the ball first rises to less than $0.5$ meters on the fifth bounce. Note: Because all the fractions have odd denominators, it is easier to double the numerators than to halve the denominators. So compare $\frac{16}{27}$ and $\frac{32}{81}$ to their numerators' fractional equivalents of $\frac{1}{2}$, $\frac{16}{32}$ and $\frac{32}{64}$.

答案（C）：下表给出了每次弹跳的高度。 \[ \begin{array}{|c|c|c|c|c|c|} \hline \text{弹跳次数} & 1 & 2 & 3 & 4 & 5\\ \hline \text{高度（米）} & 2 & \frac{2}{3}\cdot 2=\frac{4}{3} & \frac{2}{3}\cdot \frac{4}{3}=\frac{8}{9} & \frac{2}{3}\cdot \frac{8}{9}=\frac{16}{27} & \frac{2}{3}\cdot \frac{16}{27}=\frac{32}{81}\\ \hline \end{array} \] 因为 $\frac{16}{27}>\frac{16}{32}=\frac{1}{2}$ 且 $\frac{32}{81}<\frac{32}{64}=\frac{1}{2}$，所以小球第一次反弹到低于 $0.5$ 米是在第 5 次弹跳。注：由于这些分数的分母都是奇数，把分子加倍比把分母减半更容易。因此，将 $\frac{16}{27}$ 和 $\frac{32}{81}$ 与它们在“分子不变、分母取相应比较值”下的等价比较：$\frac{1}{2}$、$\frac{16}{32}$ 和 $\frac{32}{64}$ 进行比较。

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