2003 AIME II problem 4 (part 2) | Math for fun and glory | Khan Academy

TL;DR

The video explains how to find the ratio of volume between two tetrahedrons using their coordinates.

Transcript

Where we left off in the last video, we were able to figure out the coordinates of all of the vertices for this face right here of the tetrahedron. And so now we can use those coordinates to figure out the coordinates of the center of that face, which is also one of the vertices of the smaller tetrahedron. When we figure out that coordinate, then w... Read More

Key Insights

• 😀 The coordinates of the center of a tetrahedron's face can be found by averaging the coordinates of its vertices.
• 😀 The distance between the center of a face and a vertex can be used to calculate the length of a side of the smaller tetrahedron.
• 🙃 Comparing the lengths of the sides allows for determining the ratio of volume between the two tetrahedrons.

Questions & Answers

Q: How can the coordinates of the center of a tetrahedron's face be found?

To find the coordinates, you take the average of the x, y, and z coordinates of the face's vertices.

Q: How is the distance between the center and a vertex used to find the length of a side of the smaller tetrahedron?

The distance between the two points can be calculated using the Pythagorean theorem, and this distance represents the length of a side of the smaller tetrahedron.

Q: How can the ratio of volume between the two tetrahedrons be determined?

By comparing the lengths of the sides of the smaller and larger tetrahedrons, the ratio of their volumes can be found.

Q: What is the final ratio between the volumes of the tetrahedrons?

The ratio is 27 to 1, meaning the volume of the larger tetrahedron is 27 times greater than the volume of the smaller one.

Summary & Key Takeaways

• The video explains how to find the coordinates of the center of a face of a tetrahedron.

• The distance between the center and a vertex is used to find the length of a side of a smaller tetrahedron.

• By comparing the lengths of the sides of the smaller and larger tetrahedrons, the ratio of their volumes can be determined.