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Grades 11-12 Video Solutions 2023
2023_11-12_24
2023_11-12_24
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Video Transcription
24. A block in the shape of a regular tetrahedron has one face shaded. The shaded face of the block is placed on the board on the triangle labeled start. The block is then rolled from one triangle to the next by rotating about one edge. On which triangle will the block stand for the first time again on its shaded face? First step for a problem like this, always label the faces, whether that's labeling the vertices or labeling the edges. In this case, we're just going to go to directly labeling the faces. I'm just going to call the faces 1, 2, 3, and 4, where 4 is on the bottom, 3 is on the left side, 2 is on the right side, and 1 is on the back. Now, how you should solve a problem like this on the real contest is step by step, go through each of the faces, and just try to figure out the bottom face on each of the turns, so on each of the points, the start, blank space, A, B, C, D, E. Of course, since we start with the shaded block first on the bottom, we're going to label the first point 4. And then, this is where you don't actually have a tetrahedron to roll around, so you need to try to imagine what's going to happen. And for the first one, it actually doesn't matter what you pick to happen, there's a different thing that you could pick. You could have 1 being the next face that goes down, 2, or 3. As long as you're consistent throughout, you're going to get the same final answer. So, we're going to pick 2 to be the next face that goes down. If you orient a cube like this, well then the next face that's going to be on the bottom is going to be 1. And then the next face on the bottom is going to be 3. And if we keep on doing that, we'll see that after the 3, we'll rotate it again, and we'll get 2 again. And now, we're going to rotate it one more time, but this time we're not rotating it to the right, we're rotating it down, so that's going to give us 1. And then, when we rotate it to the right again, we're going to get our final answer of 4. And that means that we finally get 4 again on E. And in fact, you can try changing the 2 to 1 or 3, and you'll get the same answer regardless. And so that means that our answer is correct, and the correct answer is E.
Video Summary
To solve this problem of determining where the shaded face of a tetrahedral block will land again after rolling, follow these steps: Label the faces of the tetrahedron (1, 2, 3, 4), with 4 as the shaded bottom initially. As you roll the tetrahedron by rotating it around edges, different faces land on the bottom. Even without a physical model, imagine each step and the new bottom face. Begin with face 2, then 1, 3, and continue rotating until face 4 is again on the bottom. After several rotations, face 4 lands on triangle E. Thus, the answer is E.
Keywords
tetrahedron
rolling
shaded face
rotation
geometry
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