AMC 8 2024 Problem 3
AMC8 2024 年第 3 题
面积与周长（综合）★★☆☆☆
Four squares of side length 4, 7, 9, and 10 units are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in color white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?
将四个边长分别为 4、7、9、10 个单位的正方形从小到大依次排列，使得它们的左侧边和底边分别对齐。四个正方形的颜色按照白色-灰色-白色-灰色的顺序交替出现，如图所示。请问灰色可见区域的面积为多少个平方单位？
- A.42
- B.45
- C.49
- D.50
- E.52
正确答案 · Correct Answer
E
解析 · Solution
解法1
我们从外向内计算。外侧阴影正方形的面积等于整个正方形的面积减去第二大的正方形的面积。内侧阴影区域的面积等于第三大的正方形的面积减去最小的正方形的面积。这些面积的总和是 此题多次出现在包括AMC和MATHCOUNTS在内的各种数学竞赛中。(countmath1) 和 anabel.disher
在，我们可以使用平方差来得到答案，而不是计算这些数的平方：。
计算大正方形内每个小正方形的面积，然后从每个正方形的面积中减去白色区域的面积： 以及 anabel.disher
我们可以将其计算为2个小梯形和2个大梯形的面积之和。
Solution 1
We work inwards. The area of the outer shaded square is the area of the whole square minus the area of the second largest square. The area of the inner shaded region is the area of the third largest square minus the area of the smallest square. The sum of these areas is This problem appears multiple times in various math competitions including the AMC and MATHCOUNTS. (countmath1) and anabel.disher
In, we can use Difference of squares to get the answer, rather than calculating the squares of the numbers: .disher minor edits:)
Solve for the areas for each square inside the larger one, and then subtract the areas of the white regions from the areas of each square: and anabel.disher
We can calculate it as the sum of the areas of 2 smaller trapezoids and 2 larger trapezoids.
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