Question

Compute the volume of the following solids. Tetrahedron A tetrahedron with vertices \((0,0,0),(a, 0,0)\) \((b, c, 0),\) and \((0,0, d),\) where \(a, b, c,\) and \(d\) are positive real numbers

Short answer

Answer: The volume of the tetrahedron is $\frac{acd}{6}$.

Step-by-step solution

Set up the determinant formula for the volume of a tetrahedron

Given the vertices of the tetrahedron, we can derive a determinant formula to calculate the volume. The formula for the volume V of a tetrahedron is: $$V=\frac{1}{6}|\text{det}(A)|$$ Where A is a 3x3 matrix obtained from the coordinates of vertices of a tetrahedron: $$ A = \begin{bmatrix} a & b & 0 \\ 0 & c & 0 \\ 0 & 0 & d \end{bmatrix} $$ We will compute the determinant and then use the formula to find the volume.

Compute the determinant of matrix A

Now, calculate the determinant of the matrix A: $$\text{det}(A) = a \cdot \text{det}\begin{bmatrix} c & 0 \\ 0 & d \end{bmatrix} - b \cdot \text{det}\begin{bmatrix} 0 & 0 \\ 0 & d \end{bmatrix} + 0 \cdot \text{det}\begin{bmatrix} 0 & c \\ 0 & 0 \end{bmatrix}$$ Since the second and third terms have a row/column of zeros, they will be equal to 0. Only the first term needs to be computed: $$ \text{det}(A) = a(cd - 0) = acd $$ Now we have the determinant of matrix A, which is equal to \(acd\).

Calculate the volume of the tetrahedron

Finally, use the determinant to calculate the volume V of the tetrahedron: $$V = \frac{1}{6} |\text{det}(A)|= \frac{1}{6} |acd| = \frac{acd}{6}$$ The volume of the tetrahedron with vertices \((0,0,0)\), \((a,0,0)\), \((b,c,0)\), and \((0,0,d)\) is \(\frac{acd}{6}\).

Key concepts

Determinant Formula

The determinant formula is a mathematical tool used to compute specific values related to matrices. In the context of geometry, particularly for a three-dimensional shape like a tetrahedron, this formula plays a vital role in determining its volume. The formula for the volume of a tetrahedron, derived using determinants, is given by:

• \(V = \frac{1}{6}|\text{det}(A)|\)

The matrix \(A\) represents the geometric data obtained from the vertices of the tetrahedron. This formula allows us to calculate the volume by simply finding the determinant of the matrix formed by the coordinates of the tetrahedron's vertices.

In simpler terms, the determinant gives a scalar value which, when used in our volume formula, helps convert geometric data into an intuitive result—the volume. This method is particularly useful because it is both systematic and relies on linear algebraic principles.

Matrix Determinant

A matrix determinant is a special number that can be calculated from a square matrix. For a 3x3 matrix \(A\) used in our tetrahedron volume calculation, the determinant provides information that aids in scaling. Essentially, the determinant captures the 'scalability' of transformations applied through the matrix.

• In the matrix \(A = \begin{bmatrix} a & b & 0 \ 0 & c & 0 \ 0 & 0 & d \end{bmatrix}\), the determinant \(\text{det}(A)\) is calculated using a formula that expands according to the elements and minors of the matrix.

Calculating this involves:

• \(\text{det}(A) = a \cdot (c \cdot d) - b \cdot (0) + 0 \cdot (\ldots) = acd\)

The procedure systematically nullifies terms involving rows or columns with zeros, simplifying the calculation. Understanding this process is crucial as it links abstract algebraic operations to tangible geometric results, like volume.

3D Geometry

3D Geometry is the branch of mathematics dealing with three-dimensional forms. Tetrahedrons, such as the one we explored, are simple forms in this domain consisting of four triangular faces. Their calculations involve understanding how vertices and edges contribute to their overall shape and volume.

• The tetrahedron in this problem has vertices at \((0,0,0)\), \((a,0,0)\), \((b,c,0)\), and \((0,0,d)\).

The volume computation uses how these vertices are positioned to determine how much space the shape occupies. The matrix and its determinant capture the relationship between these vertices, translating into the volume formula.

• Understanding such relationships in 3D Geometry helps in visualizing complex structures and solving related problems across various fields.

Calculus

Calculus is a field of mathematics that focuses on change and motion. While it may not appear directly in the simplified formula for tetrahedron volume, underlying concepts from calculus, like integration, often aid in comprehending and deriving such geometric formulas.

• Using determinants is akin to integrating discrete changes across the axes defined by the matrix.

In essence, calculus provides tools for rigorously defining and understanding continuous changes and areas under curves—concepts that neatly intertwine with the principles behind volume and surface calculations.

• For students familiar with calculus, seeing these connections can deepen their understanding of why particular formulas in geometry work as they do.

Understanding these concepts can illuminate the processes and logic underpinning more straightforward geometric computations.