Question

七个立方体的体积分别为 $1$ 、 $8$ 、 $27$ 、 $64$ 、 $125$ 、 $216$ 和 $343$ 立方单位。它们竖直堆成一座塔，体积从底部到顶部递减。除最底部的立方体外，每个立方体的底面都完全位于其下方立方体的顶面上。求这座塔的总表面积，包括底面。

Seven cubes, whose volumes are $1,$ $8,$ $27,$ $64,$ $125,$ $216,$ and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

$644$

$658$

$664$

$720$

$749$

答案：B

知识点：表面积正方体前n项平方和

难度评级：1420

视频讲解：

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文字解答：

这些立方体边长为 $1,2,3,4,5,6,7$ ，从大到小堆叠。单独表面积之和为 $6(1^2+2^2+\cdots+7^2)$ $=6\cdot140=840$ 。

每个接触处隐藏两个正方形面，面积分别为 $1^2,2^2,\ldots,6^2$ 。所以总表面积为 $840$ $-2(1^2+2^2+\cdots+6^2)$ $=840-182=658$ 。正确答案是 B。

The cube side lengths are $1,2,3,4,5,6,7$ , stacked from largest on bottom to smallest on top. The sum of the surface areas of the separate cubes is $6(1^2+2^2+\cdots+7^2)$ $=6\cdot140=840$ .

Each contact hides two square faces, with areas $1^2,2^2,\ldots,6^2$ . Subtracting these hidden faces gives $840$ $-2(1^2+2^2+\cdots+6^2)$ $=840-182=658$ . Thus, B is the correct answer.

2020 AMC 10A 第 10 题

答案：B

难度评级：1420

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文字解答：

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