Question

一块地毯由如图所示的三种不同颜色组成。三种不同颜色区域的面积构成一个等差数列。最内层长方形宽为一英尺，外面的两个区域在四周各宽 $1$ 英尺。最内层长方形的长度是多少英尺？

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?

$1$

$2$

$4$

$6$

$8$

答案：B

知识点：面积等差数列

难度评级：1370

解答：

设内层长方形长度为 $l$ ，则内层面积为 $l$ 。

中间区域面积为外层区域面积为 $(l + 2) \cdot 3 - l = 2l + 6.$ $(l + 4) \cdot 5 - (l + 2) \cdot 3 = 2l + 14.$

这 $3$ 个面积构成等差数列，所以 $\begin{aligned} l + 2l + 14 &= 2(2l + 6) \\ 3l + 14 &= 4l + 12 \\ l &= 2\end{aligned}$

所以正确答案是 B。

Let $l$ be the length of the inner rectangle. Then the area of the inner rectangle is $l.$

The area of the middle region is going to be $(l + 2) \cdot 3 - l = 2l + 6.$ The area of the outer region is $(l + 4) \cdot 5 - (l + 2) \cdot 3 = 2l + 14.$

We know that these $3$ values form an arithmetic sequence. That means that $\begin{aligned} l + 2l + 14 &= 2(2l + 6) \\ 3l + 14 &= 4l + 12 \\ l &= 2\end{aligned}$

Thus, the correct answer is B .

2016 AMC 10A 第 10 题

答案：B

难度评级：1370

解答：

其他年份的第 10 题