AMC8 2020

AMC8 2020 · Q20

AMC8 2020 · Q20. It mainly tests Word problems (algebra), Averages (mean).

A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?

一位科学家在森林中行走，记录了 $5$ 棵成排站立的树的整数高度。她观察到每棵树要么是其右侧树的两倍高，要么是其一半高。不幸的是，她的笔记本被雨淋湿，有些数据丢失了。她的笔记如下，空白处表示缺失的数字。根据她的观察，科学家能够重建丢失的数据。树的平均高度是多少米？

(A) 22.2 22.2

(B) 24.2 24.2

(D) 35.2 35.2

(E) 37.2 37.2

Answer

Correct choice: (B)

正确答案：（B）

Solution

We will show that $22$, $11$, $22$, $44$, and $22$ meters are the heights of the trees from left to right. We are given that all tree heights are integers, so since Tree 2 has height $11$ meters, we can deduce that Trees 1 and 3 both have a height of $22$ meters. There are now three possible cases for the heights of Trees 4 and 5 (in order for them to be integers), namely heights of $11$ and $22$, $44$ and $88$, or $44$ and $22$. Checking each of these, in the first case, the average is $17.6$ meters, which doesn't end in $.2$ as the problem requires. Therefore, we consider the other cases. With $44$ and $88$, the average is $37.4$ meters, which again does not end in $.2$, but with $44$ and $22$, the average is $24.2$ meters, which does. Consequently, the answer is $\boxed{\textbf{(B) }24.2}$.

我们将证明树的从左到右高度分别是 $22$、$11$、$22$、$44$ 和 $22$ 米。已知所有树高为整数，由于第 $2$ 棵树高度 $11$ 米，我们可以推断第 $1$ 和第 $3$ 棵树高度均为 $22$ 米。现在第 $4$ 和第 $5$ 棵树有三种可能整数高度情况，即 $11$ 和 $22$、$44$ 和 $88$，或 $44$ 和 $22$。检查每种情况，第一种平均高度 $17.6$ 米，不以 $.2$ 结尾，不符合题目要求。因此考虑其他情况。$44$ 和 $88$ 时平均 $37.4$ 米，也不以 $.2$ 结尾，但 $44$ 和 $22$ 时平均 $24.2$ 米，符合。因此答案是 $\boxed{\textbf{(B) }24.2}$。

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