AMC 8 2013 Problem 18

AMC8 2013 年第 18 题

立体几何★★★☆☆

Isabella uses one-foot cubical blocks to build a rectangular fort that is 12 feet long, 10 feet wide, and 5 feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

Isabella 使用一些边长为 1 英尺的正方体砖块来建造一个长为 12 英尺，宽 10 英尺，高 5 英尺的堡垒。堡垒的地板和 4 堵墙的厚度都是 1 英尺。那么这个堡垒需要多少块砖块？

A.204
B.280
C.320
D.340
E.600

正确答案 · Correct Answer

解析 · Solution

Solution 1

There are

10 \cdot 12 = 120

cubes on the base of the box. Then, for each of the 4 layers above the bottom (as since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there are 9 + 11 + 9 + 11 = 40 cubes. Hence, the answer is

120 + 4 \cdot 40 = 280

. We can just calculate the volume of the prism that was cut out of the original 12×10×5 box. Each interior side of the fort will be 2 feet shorter than each side of the outside. Since the floor is 1 foot, the height will be 4 feet. So the volume of the interior box is

10×8\times 4=320\text{ ft}^3

. The volume of the original box is

12×10×5=600\text{ ft}^3

. Therefore, the number of blocks contained in the fort is 600-320=280

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