Math Problem Statement

Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.)

Solution

To solve this problem, we need to find the volume of the given right tetrahedron. A tetrahedron is a three-dimensional figure with four triangular faces. The formula for the volume VV of a tetrahedron with a right-angle vertex can be expressed as:

V=13Base AreaHeightV = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height}

In the provided diagram, the tetrahedron has the right-angle vertex at point zz, with three labeled edges forming a right triangle. These labeled edges provide the base and height for our calculations.

The base triangle lies in the xyxy-plane, where:

  • The edge along the xx-axis is 100 units.
  • The edge along the yy-axis is 96 units.

These two edges form a right triangle, so the area of this triangular base is:

Base Area=1210096\text{Base Area} = \frac{1}{2} \cdot 100 \cdot 96

The height of the tetrahedron is along the zz-axis, which is 128 units.

Now, we can calculate the volume:

= \frac{1}{3} \cdot \frac{1}{2} \cdot 100 \cdot 96 \cdot 128$$ Let's compute this.The volume of the given right tetrahedron is $$204800$$ cubic units. Would you like a detailed breakdown of the steps, or do you have any further questions? Here are 5 related questions you may find helpful: 1. How do you calculate the volume of a general tetrahedron? 2. What are the properties of a right tetrahedron? 3. How is the volume of a pyramid related to the volume of a tetrahedron? 4. How can you use vectors to find the volume of a tetrahedron? 5. How do you find the surface area of a tetrahedron? **Tip:** Always make sure that the units for all dimensions are consistent before calculating volume or area in any problem!

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Math Problem Analysis

Mathematical Concepts

Geometry
Solid Figures
Tetrahedron Volume

Formulas

Volume of a tetrahedron: V = (1/3) * Base Area * Height
Base Area of a triangle: A = (1/2) * base * height

Theorems

Volume of Pyramids and Tetrahedrons

Suitable Grade Level

Grades 9-12