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| − | ==Problem==
| + | #REDIRECT [[2019 MPFG Problem 17]] |
| − | Let <math>P</math> be a right prism whose two bases are equilateral triangles with side length <math>2</math>. The height of <math>P</math> is <math>2\sqrt{3}</math>. Let <math>l</math> be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases <math>180^\circ</math> about <math>l</math>. Let <math>T</math> be the convex hull of the two current triangles. What is the volume of <math>T</math>?
| + | <br> |
| − | | + | {{delete|housekeeping}} |
| − | ==Solution 1==
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| − | Here is a demonstration of the actual transformation
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| − | [[File:2019MPFG_17.jpg|450px|center]] | |
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| − | As we can see, the transformation creates a rectangular prism with <math>4</math> triangular pyramids cut off from the corners.
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| − | The volume of the rectangular prism is
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| − | <cmath> 2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12</cmath>
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| − | Subtract the volume of the <math>4</math> triangular pyramids, and we get:
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| − | <cmath>V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}</cmath>
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| − | <cmath>= 12 - 8 = \boxed{4}</cmath>
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| − | ~cassphe
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