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Solution 2
We can use Principle of Inclusion-Exclusion to find the final volume of the cube. There are 3 \\
volumes, as the central
cubic inches. However, we can not just sum their
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 .
Solution 3
We can visualize the final figure and see a cubic frame. We can find the volume of the figure by adding up the volumes of the edges and corners. Each edge can be seen as a
box.
box, and each corner can be seen as a
.
Solution 4
First, you can find the volume, which is 27. Now, imagine there are three prisms of dimensions 2 x 2 x 3. Now subtract the prism volumes from 27. We have -9. From here we add two times 2^3, because we over-removed (LOL). This is 16 - 9 = 7 (A).
Problem 10
The first four terms of an arithmetic sequence are , , is the
term of this sequence?
, and
. What
Solution
and
are consecutive terms, so the common difference is
.
The common difference is . The first term is and the
term is
Problem 11
The solution of the equation What is ?
can be expressed in the form
.
Solution
This problem is quickly solved with knowledge of the laws of exponents and logarithms.
Since we are looking for the base of the logarithm, our answer is .
Problem 12
In a magical swamp