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Difference between revisions of "2019 MPFG Problems/Problem 17"
(→‎Solution 1)
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==Solution 1==
 
==Solution 1==
−
Here is a demonstration of
+
Here is a demonstration of the actual transformation
 
[[File:2019MPFG_17.jpg|450px]]
 
[[File:2019MPFG_17.jpg|450px]]
  
−
As we can see, the transformation creates a rectangular prism with <math>4</math> triangular cut-off from the corners.
+
As we can see, the transformation creates a rectangular prism with <math>4</math> triangular pyramids cut off from the corners.
  
−
The volume of the rectangular prism  
+
The volume of the rectangular prism is
 
<cmath> 2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12</cmath>
 
<cmath> 2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12</cmath>
  
−
Subtract the volume of the <math>4</math> triangular prisms, and we get:
+
Subtract the volume of the <math>4</math> triangular pyramids, and we get:
 
<cmath>V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}</cmath>
 
<cmath>V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}</cmath>
 
<cmath>= 12 - 8 = \boxed{4}</cmath>
 
<cmath>= 12 - 8 = \boxed{4}</cmath>

Revision as of 08:55, 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 the actual transformation 2019MPFG 17.jpg

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

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

Subtract the volume of the $4$ triangular pyramids, 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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