Question

一架飞机逆风直飞两座城镇之间，耗时 $84$ 分钟；顺风返回所需时间比无风时少 $9$ 分钟。返程所需的分钟数（有两个答案）为

A plane flew straight against a wind between two towns in $84$ minutes and returned with that wind in $9$ minutes less than it would take in still air. The number of minutes (two answers) for the return trip was

$54$ 或 $18$

$54$ or $18$

$60$ 或 $15$

$60$ or $15$

$63$ 或 $12$

$63$ or $12$

$72$ 或 $36$

$72$ or $36$

$75$ 或 $20$

$75$ or $20$

答案：C

知识点：路程、速度与时间二次方程方程组

难度评级：2410

小提示：

设返程时间为 $x$ ；则无风时的飞行时间为 $x+9$

Let $x$ be the return time; the still-air time is then $x+9$

大提示：

飞机在无风时的速度是逆风与顺风地速的平均值

The plane’s still-air speed is the average of its against-wind and with-wind ground speeds

解答：

设两城距离为 $d$ ，返程时间为 $x$ 分钟。逆风与顺风速度分别为 $\frac{d}{84}$ 与 $\frac{d}{x}$ 。它们的平均值是无风速度 $\frac{d}{x+9}$ 。因此 $\frac{2}{x+9}=\frac1{84}+\frac1x\text{。}$ 清除分母，得到 $\begin{gathered} 168x=(x+9)(x+84),\\ x^2-75x+756=0,\\ (x-63)(x-12)=0 \end{gathered}\text{。}$ 两个正值都符合题设条件，所以两个返程时间为 $63$ 分钟与 $12$ 分钟。

所以正确答案是 C。

Let the distance be $d$ and the return time be $x$ minutes. The against-wind and with-wind speeds are $\frac{d}{84}$ and $\frac{d}{x}.$ Their average is the still-air speed $\frac{d}{x+9}.$ Therefore $\frac{2}{x+9}=\frac1{84}+\frac1x.$ Clearing denominators gives $\begin{gathered} 168x=(x+9)(x+84),\\ x^2-75x+756=0,\\ (x-63)(x-12)=0. \end{gathered}$ Both positive values are consistent with the stated conditions, so the two return times are $63$ and $12$ minutes.

Therefore, the correct answer is C.

1973 AMC 12 第 34 题

答案：C

难度评级：2410

小提示：

大提示：

解答：

其他年份的第 34 题