Question

一个 $2 \times 3$ 矩形和一个 $3 \times 4$ 矩形放在一个正方形内，内部不重叠，且正方形的边与两个矩形的边平行。这个正方形最小可能面积是多少？

A $2 \times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?

$16$

$25$

$36$

$49$

$64$

答案：B

知识点：正方形（几何）矩形最优化

难度评级：1060

小提示：

试着把两个矩形并排放，使它们长度为 $3$ 的边对齐。

Try stacking the rectangles so their sides of length $3$ line up

大提示：

正方形边长至少要容纳两个较小尺寸之和 $2+3$ 。

The square’s side must be at least the sum of the smaller dimensions, $2+3$

解答：

将两个矩形并排放置，使长度为 $3$ 的边竖直。它们宽度相加为 $2+3=5$ ，高度 $3$ 和 $4$ 都能放进边长 $5$ 的正方形。

因为两个矩形都与正方形的边平行且内部不重叠，所以它们的水平投影或竖直投影必定互不重叠。在任一方向上，第一个矩形至少占据 $2$ 的长度，第二个矩形至少占据 $3$ 的长度，因此正方形边长至少为 $2+3=5$ 。所以最小面积为 $5^2=25$ 。

所以正确答案是 B。

Place the rectangles side by side with their $3$ -length sides vertical. Their widths add to $2+3=5,$ and the heights $3$ and $4$ both fit within $5.$

Because the rectangles are axis-aligned and their interiors do not overlap, their horizontal projections or their vertical projections must be disjoint. In either direction, the first rectangle spans at least $2$ and the second spans at least $3,$ so the square’s side is at least $2+3=5.$ The smallest area is therefore $5^2=25.$

Thus, the correct answer is B.

2006 AMC 10B 第 5 题

答案：B

难度评级：1060

小提示：

大提示：

解答：

其他年份的第 5 题