Rubik’s Tetrahedron Rubik’s Cube, a puzzle craze of the 1980s that remains popular to this day, inspired many similar puzzles. The one illustrated in the figure is called Rubik’s Tetrahedron; it is in the shape of a regular tetrahedron, with each edge
- (a) Use the triple product to find the volume of Rubik’s Tetrahedron. [Hint: See Exercise 50 in Section 9.4, which gives the corners of a tetrahedron that has the same shape and size as Rubik’s Tetrahedron.]
- (b) Construct six identical regular tetrahedra using modeling clay. Experiment to see how they can be put together to create a parallelepiped that is determined by three edges of one of the tetrahedra (thus confirming the above statement about the volume of a regular tetrahedron).
The volume of Rubik’s Tetrahedron with the use of triple product.
Answer to Problem 36E
The volume of Rubik’s Tetrahedron is
Explanation of Solution
Given:
A Rubik’s Tetrahedron with edge
Formula used:
The volume of regular tetrahedron is
Volume of Parallelepiped is
Calculation:
Each edge of the tetrahedron is
Find the magnitude of the vector
The angle between the two edges of a regular tetrahedron is
Substitute
The direction of
Evaluate the magnitude
The volume of regular tetrahedron is
Thus, the volume of Rubik’s tetrahedron is
Chapter 9 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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