Question

一个以正方形为底的封闭盒子，要用一张正方形包装纸包装。盒子居中放在包装纸上，底面顶点落在正方形纸的中线上，如左图所示。包装纸四个角将沿侧面向上折起，并在盒子顶部中心点 $A$ 处相遇，如右图所示。盒子底边长为 $w$ ，高为 $h$ 。包装纸的面积是多少？

A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h.$ What is the area of the sheet of wrapping paper?

$2(w+h)^2$

$\dfrac{(w+h)^2}{2}$

$2w^2+4wh$

$2w^2$

$w^2h$

答案：A

知识点：特殊直角三角形面积

难度评级：1760

小提示：

从包装纸一角到纸中心的距离为 $\tfrac{w}{2}+h+\tfrac{w}{2}$ 。

The distance from a corner of the sheet to its center is $\tfrac{w}{2}+h+\tfrac{w}{2}$

大提示：

正方形包装纸的边长是这段角到中心距离的 $\sqrt2$ 倍。

A side of the square sheet is $\sqrt2$ times that corner-to-center distance

解答：

沿着折线从纸角到盒顶中心看，纸角到纸中心的距离为 $\dfrac{w}{2}+h+\dfrac{w}{2}=w+h\text{。}$

这段是 $45$ - $45$ - $90$ 三角形的一条直角边，其斜边是正方形包装纸的一整条边，所以边长为 $\sqrt2\,(w+h)$ 。

包装纸面积为 $\left(\sqrt2\,(w+h)\right)^2=2(w+h)^2$ 。

所以正确答案是 A。

Following a fold from a corner of the paper to the center of the box top, the distance from a corner of the sheet to its center is $\dfrac{w}{2}+h+\dfrac{w}{2}=w+h.$

That segment is a leg of a $45$ - $45$ - $90$ triangle whose hypotenuse is a full side of the square sheet, so the side length is $\sqrt2\,(w+h).$

The area of the sheet is $\left(\sqrt2\,(w+h)\right)^2=2(w+h)^2.$

Thus, the correct answer is A.

2018 AMC 12B 第 11 题

答案：A

难度评级：1760

小提示：

大提示：

解答：

其他年份的第 11 题