AMC10 2017 B

AMC10 2017 B · Q6

AMC10 2017 B · Q6. It mainly tests 3D geometry (volume), 3D geometry (surface area).

What is the largest number of solid 2-in × 2-in × 1-in blocks that can fit in a 3-in × 2-in × 3-in box?

一个 $3$-英寸 $ imes 2$-英寸 $ imes 3$-英寸的盒子中，能放入的最大个数 $2$-英寸 $ imes 2$-英寸 $ imes 1$-英寸的实心方块是多少？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (B)

正确答案：（B）

Solution

A possible arrangement of 4 blocks is shown by the figure. Four blocks do not completely fill the box because the combined volume of the blocks is only 4(2 · 2 · 1) = 16 cubic inches, whereas the volume of the box is 3 · 2 · 3 = 18 cubic inches. Because the unused space, 18 − 16 = 2 cubic inches, is less than the volume of a block, 4 cubic inches, no more than 4 blocks can fit in the box.

图中展示了一种放置 4 个方块的可能排列。四个方块不能完全填满盒子，因为方块的总体积只有 $4(2 \cdot 2 \cdot 1)=16$ 立方英寸，而盒子体积是 $3 \cdot 2 \cdot 3=18$ 立方英寸。由于剩余空间 $18-16=2$ 立方英寸小于一个方块的体积 4 立方英寸，因此最多只能放入 4 个方块。

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