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"

(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...")
 
m (→‎Solution 1)
 
(2 intermediate revisions by the same user not shown)
Line 3: Line 3:
  
 
==Solution 1==
 
==Solution 1==
 +
Here is a demonstration of the actual transformation
 +
[[File:2019MPFG_17.jpg|450px]]
 +
 +
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 is
 +
<cmath> 2 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3} = 12</cmath>
 +
 +
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>= 12 - 8 = \boxed{4}</cmath>
 +
 +
~cassphe

Latest revision as of 09:30, 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: \[V = 12 - 4 \cdot \frac{1}{2} \cdot 1 \cdot (2 \cdot \frac{\sqrt{3}}{2}) \cdot 2\sqrt{3}\] \[= 12 - 8 = \boxed{4}\]

~cassphe

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_MPFG_Problems/Problem_17&oldid=257050"