AMC12 2003 A

AMC12 2003 A · Q3

AMC12 2003 A · Q3. It mainly tests 3D geometry (volume), 3D geometry (surface area).

A solid box is $15$ cm by $10$ cm by $8$ cm. A new solid is formed by removing a cube $3$ cm on a side from each corner of this box. What percent of the original volume is removed?

一个实心长方体盒子的尺寸为 $15$ cm $\times 10$ cm $\times 8$ cm。从这个盒子的每个角上都切去一个边长为 $3$ cm 的立方体，形成一个新的实心体。被切去的体积占原体积的百分之多少？

(A) 4.5 4.5

(B) 9 9

(D) 18 18

(E) 24 24

Answer

Correct choice: (D)

正确答案：（D）

Solution

The volume of the original box is $15\cdot10\cdot8=1200.$ The volume of each cube that is removed is $3\cdot3\cdot3=27.$ Since there are $8$ corners on the box, $8$ cubes are removed. So the total volume removed is $8\cdot27=216$. Therefore, the desired percentage is $\frac{216}{1200}\cdot100 = \boxed{\mathrm{(D)}\ 18\%}.$

原盒子的体积为 $15\cdot10\cdot8=1200.$ 每个被切去的立方体体积为 $3\cdot3\cdot3=27.$ 由于盒子有 $8$ 个角，所以共切去 $8$ 个立方体。因此被切去的总体积为 $8\cdot27=216$。所求百分比为 $\frac{216}{1200}\cdot100 = \boxed{\mathrm{(D)}\ 18\%}.$

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