### Calculating the Volume of a Tetrahedron**Problem Statement:**1) Compute the volume of a tetrahedron.**Step-by-Step Solution:**(a) Illustrate the tetrahedron that has vertices at (0,0,0), (2,0,0), (0,4,0), and (0,0,6) in Cartesian coordinates.This tetrahedron sits inside a rectangular box with side lengths 2, 4, and 6. The volume of this box is calculated using the formula for the volume of a rectangular prism:\[ V_{\text{box}} = \text{length} \times \text{width} \times \text{height} \]Substituting the given dimensions:\[ V_{\text{box}} = 2 \times 4 \times 6 = 48 \text{ cubic units} \]Since the tetrahedron occupies a portion of this box, the volume of the tetrahedron can be found by determining what fraction of the box it occupies. The vertices of the tetrahedron suggest it is a specific fraction of the box:\[ V_{\text{tetrahedron}} = \frac{V_{\text{box}}}{6} \]This is because a tetrahedron that fits perfectly inside a rectangular box defined by one vertex at the origin and other vertices at the axes will occupy 1/6th of the volume of the box.Therefore, the volume of the tetrahedron is:\[ V_{\text{tetrahedron}} = \frac{48}{6} = 8 \text{ cubic units} \]**Summary:**- The volume of the tetrahedron with the given vertices is **8 cubic units**.

Tetrahedron can also be seen as triangular pyramid, so the volume of tetrahedron is equal to the…

Compute the volume of a tetrahedron. (a) Illustrate the tetrahedron that has vertices at (0,0,0), (2,0,0), (0, 4, 0), (0,0, 6), in Cartesian coordinates. This tetrahedron sits inside a box with side lengths 2, 4 and 6. The volume of this box is V = 2 x 4 x 6 = 48 cubic units. The volume of the tetrahedron must be some fraction of this.

Compute the volume of a tetrahedron. (a) Illustrate the tetrahedron that has vertices at (0,0,0), (2,0,0), (0, 4, 0), (0,0, 6), in Cartesian coordinates. This tetrahedron sits inside a box with side lengths 2, 4 and 6. The volume of this box is V = 2 x 4 x 6 = 48 cubic units. The volume of the tetrahedron must be some fraction of this.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

### Calculating the Volume of a Tetrahedron

**Problem Statement:**
1) Compute the volume of a tetrahedron.

**Step-by-Step Solution:**
(a) Illustrate the tetrahedron that has vertices at (0,0,0), (2,0,0), (0,4,0), and (0,0,6) in Cartesian coordinates.

This tetrahedron sits inside a rectangular box with side lengths 2, 4, and 6. The volume of this box is calculated using the formula for the volume of a rectangular prism:

\[ V_{\text{box}} = \text{length} \times \text{width} \times \text{height} \]

Substituting the given dimensions:
\[ V_{\text{box}} = 2 \times 4 \times 6 = 48 \text{ cubic units} \]

Since the tetrahedron occupies a portion of this box, the volume of the tetrahedron can be found by determining what fraction of the box it occupies. The vertices of the tetrahedron suggest it is a specific fraction of the box:

\[ V_{\text{tetrahedron}} = \frac{V_{\text{box}}}{6} \]

This is because a tetrahedron that fits perfectly inside a rectangular box defined by one vertex at the origin and other vertices at the axes will occupy 1/6th of the volume of the box.

Therefore, the volume of the tetrahedron is:
\[ V_{\text{tetrahedron}} = \frac{48}{6} = 8 \text{ cubic units} \]

**Summary:**
- The volume of the tetrahedron with the given vertices is **8 cubic units**.

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