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Difference between revisions of "2019 MPFG Problems/Problem 17"

(Created page with "==Problem== 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}</mat...")
 
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==Solution 1==
 
==Solution 1==
 +
Here is a demonstration of
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[[File:2019MPFG_17.jpg|450px]]
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 +
As we can see, the transformation creates a rectangular prism with <math>4</math> triangular cut-off from the corners.
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The volume of the rectangular prism
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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 prisms, 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>

Revision as of 08:54, 11 August 2025

Problem

Let $P$ be a right prism whose two bases are equilateral triangles with side length $2$. The height of $P$ is $2\sqrt{3}$. Let l be the line connecting the centroids of the bases. Remove the solid, keeping only the bases. Rotate one of the bases $180\circ$ about l. Let $T$ be the convex hull of the two current triangles. What is the volume of $T$?

Solution 1

Here is a demonstration of 2019MPFG 17.jpg

As we can see, the transformation creates a rectangular prism with $4$ triangular cut-off from the corners.

The volume of the rectangular prism \[2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12\]

Subtract the volume of the $4$ triangular prisms, and we get: \[V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}\] \[= 12 - 8 = \boxed{4}\]

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