Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎Solution 1)
 
Line 1: Line 1:
 
==Problem==
 
==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}</math>. Let l 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 l. Let <math>T</math> be the convex hull of the two current triangles. What is the volume of <math>T</math>?
+
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>?
  
 
==Solution 1==
 
==Solution 1==
 
Here is a demonstration of the actual transformation
 
Here is a demonstration of the actual transformation
−
[[File:2019MPFG_17.jpg|450px]]
+
 
 +
[[File:2019MPFG_17.jpg|450px|center]]
  
 
As we can see, the transformation creates a rectangular prism with <math>4</math> triangular pyramids 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.

Latest revision as of 19:36, 16 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: \[V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}\] \[= 12 - 8 = \boxed{4}\]

~cassphe

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_MPFG_Problems/Problem_17&oldid=257875"