Difference between revisions of "2010 AMC 12A Problems/Problem 9"
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== Solution == | == Solution == | ||
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| + | === Solution 1 === | ||
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| + | Imagine making the cuts one at a time. The first cut removes a box <math>2\times 2\times 3</math>. The second cut removes two boxes, each of dimensions <math>2\times 2\times 0.5</math>, and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is <math>12 + 4 + 4 = 20</math>. | ||
| + | |||
| + | === Solution 2 === | ||
| + | |||
We can use [[Principle of Inclusion-Exclusion]] to find the final volume of the cube. | We can use [[Principle of Inclusion-Exclusion]] to find the final volume of the cube. | ||
| − | There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" | + | There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has <math>2 \times 2 \times 3=12</math> cubic inches. However, we can not just sum their volumes, as |
| + | the central <math>2\times 2\times 2</math> cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice. | ||
| + | Hence the total volume of the cuts is <math>3(2 \times 2 \times 3) - 2(2\times 2\times 2) = 36 - 16 = 20</math>. | ||
| − | <math> | + | Therefore the volume of the rest of the cube is <math>3^3 - 20 = 27 - 20 = \boxed{7\ \textbf{(A)}}</math>. |
Revision as of 10:24, 16 February 2010
Problem 9
A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
Solution
Solution 1
Imagine making the cuts one at a time. The first cut removes a box 
. The second cut removes two boxes, each of dimensions 
, and the third cut does the same as the second cut, on the last two faces. Hence the total volume of all cuts is 
.
Solution 2
We can use Principle of Inclusion-Exclusion to find the final volume of the cube.
There are 3 "cuts" through the cube that go from one end to the other. Each of these "cuts" has 
cubic inches. However, we can not just sum their volumes, as
the central 
cube is included in each of these three cuts. To get the correct result, we can take the sum of the volumes of the three cuts, and subtract the volume of the central cube twice.
Hence the total volume of the cuts is 
.
Therefore the volume of the rest of the cube is 
.
