AMC10 2016 A

AMC10 2016 A · Q10

AMC10 2016 A · Q10. It mainly tests Vieta / quadratic relationships (basic), Area & perimeter.

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?

如图所示，一块地毯由三种不同颜色组成。三块不同颜色区域的面积成等差数列。内层矩形的宽为 1 英尺，且两层阴影区域在四个方向的宽度都为 1 英尺。问内层矩形的长是多少英尺？

(A) 1 1

(B) 2 2

(D) 6 6

(E) 8 8

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let the inner rectangle’s length be $x$ feet; then its area is $x$ square feet. The middle region has area $3(x+2)-x=2x+6$, so the difference in the arithmetic sequence is equal to $(2x+6)-x=x+6$. The outer region has area $5(x+4)-3(x+2)=2x+14$, so the difference in the arithmetic sequence is also equal to $(2x+14)-(2x+6)=8$. From $x+6=8$, it follows that $x=2$. The regions have areas $2$, $10$, and $18$.

答案（B）：设内矩形的长度为 $x$ 英尺，则其面积为 $x$ 平方英尺。中间区域的面积为 $3(x+2)-x=2x+6$，因此等差数列的公差为 $(2x+6)-x=x+6$。外部区域的面积为 $5(x+4)-3(x+2)=2x+14$，因此等差数列的公差也等于 $(2x+14)-(2x+6)=8$。由 $x+6=8$ 得 $x=2$。这些区域的面积分别为 $2$、$10$ 和 $18$。

Topics