AMC12 2013 B

AMC12 2013 B · Q11

AMC12 2013 B · Q11. It mainly tests Coordinate geometry, Distance / midpoint.

Two bees start at the same spot and fly at the same rate in the following directions. Bee A travels 1 foot north, then 1 foot east, then 1 foot upwards, and then continues to repeat this pattern. Bee B travels 1 foot south, then 1 foot west, and then continues to repeat this pattern. In what directions are the bees traveling when they are exactly 10 feet away from each other?

两只蜜蜂从同一位置开始，以相同的速度向以下方向飞行。蜜蜂A向北飞行1英尺，然后向东1英尺，然后向上1英尺，然后继续重复此模式。蜜蜂B向南飞行1英尺，然后向西1英尺，然后继续重复此模式。当它们彼此相距恰好10英尺时，它们正在向哪些方向飞行？

(A) A east, B west A 向东，B 向西

(B) A north, B south A 向北，B 向南

(D) A up, B south A 向上，B 向南

(E) A up, B west A 向上，B 向西

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): Suppose that the two bees start at the origin and that the positive directions of the $x$, $y$, and $z$ coordinate axes correspond to the directions east, north, and up, respectively. Note that the bees are always getting farther apart from each other. After bee $A$ has traveled 7 feet it will have gone 3 feet north, 2 feet east, and 2 feet up. Its position would be the point $(2,3,2)$. In the same time bee $B$ will have gone 4 feet south and 3 feet west, and its position would be the point $(-3,-4,0)$. This puts them at a distance $$ \sqrt{(2-(-3))^2+(3-(-4))^2+2^2}=\sqrt{78}<10 $$ After this moment, bee $A$ will travel east to the point $(3,3,2)$ and bee $B$ will travel west to the point $(-4,-4,0)$. Their distance after traveling one foot will be $$ \sqrt{(3-(-4))^2+(3-(-4))^2+2^2}=\sqrt{102}>10. $$ Hence bee $A$ is traveling east and bee $B$ is traveling west when they are exactly 10 feet away from each other.

答案（A）：设两只蜜蜂从原点出发，并规定坐标轴 $x$、$y$、$z$ 的正方向分别对应向东、向北和向上。注意两只蜜蜂彼此之间的距离始终在增大。当蜜蜂 $A$ 飞行了 7 英尺时，它将向北 3 英尺、向东 2 英尺、向上 2 英尺，因此其位置为点 $(2,3,2)$。在同一时间内，蜜蜂 $B$ 将向南 4 英尺、向西 3 英尺，其位置为点 $(-3,-4,0)$。此时它们之间的距离为 $$ \sqrt{(2-(-3))^2+(3-(-4))^2+2^2}=\sqrt{78}<10 $$ 在此之后，蜜蜂 $A$ 将向东飞到点 $(3,3,2)$，蜜蜂 $B$ 将向西飞到点 $(-4,-4,0)$。它们各再飞行 1 英尺后的距离为 $$ \sqrt{(3-(-4))^2+(3-(-4))^2+2^2}=\sqrt{102}>10. $$ 因此，当它们之间的距离恰好为 10 英尺时，蜜蜂 $A$ 正向东飞行，而蜜蜂 $B$ 正向西飞行。

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