AMC10 2019 B

AMC10 2019 B · Q18

AMC10 2019 B · Q18. It mainly tests Functions basics, Patterns & sequences (misc).

Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A - B|$?

亨利某天早上决定锻炼，他从家走到健身房途中的 $\frac{3}{4}$ 距离。健身房离亨利家 2 千米。在那个点，他改变主意，从当前位置向家走 $\frac{3}{4}$ 的距离。当他到达那个点时，又改变主意，从那里向健身房走 $\frac{3}{4}$ 的距离。如果亨利每次在走完从上次改变主意点向健身房或家 $\frac{3}{4}$ 距离时改变主意，他将非常接近在离家 $A$ 千米和离家 $B$ 千米的两个点之间来回走。$|A - B|$ 是多少？

(A) $\frac{2}{3}$ $\frac{2}{3}$

(B) $1$ $1$

(D) $\frac{1}{4}$ $\frac{1}{4}$

(E) $\frac{1}{2}$ $\frac{1}{2}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): By symmetry Henry’s walks will converge toward walking between two points, one at a distance $x$ from the gym and the other at the same distance $x$ from his home. Because Henry would be $2-x$ kilometers from home when he is closest to the gym and also because his trip toward home would take him to $\frac{1}{4}$ this distance from home, $x=\frac{1}{4}(2-x)$. Solving this yields $x=\frac{2}{5}$. Therefore, Henry’s walks will approach $2-2\cdot\frac{2}{5}=1\frac{1}{5}$ kilometers in length.

答案（C）：由对称性，亨利的步行将收敛为在两点之间来回走动：一点距离健身房为 $x$，另一点距离他家也为 $x$。因为当他离健身房最近时，他离家为 $2-x$ 千米；并且他朝家的这段行程会使他到达距离家为该距离的 $\frac{1}{4}$，所以有 $x=\frac{1}{4}(2-x)$。解得 $x=\frac{2}{5}$。因此，亨利的步行长度将趋近于 $2-2\cdot\frac{2}{5}=1\frac{1}{5}$ 千米。

Topics