Problem 36
Question
Write inequalities to describe the sets in Exercises \(35-40\) The solid cube in the first octant bounded by the coordinate planes and the planes \(x=2, y=2,\) and \(z=2\)
Step-by-Step Solution
Verified Answer
The cube is described by the inequalities: \(0 \leq x \leq 2\), \(0 \leq y \leq 2\), \(0 \leq z \leq 2\).
1Step 1: Understand the Problem
The problem describes a solid cube that is in the first octant. This means that the cube is bounded by the coordinate planes (where the x, y, and z coordinates are non-negative) and the planes defined by the equations \(x=2\), \(y=2\), and \(z=2\). These planes form the boundaries of the cube.
2Step 2: Identify the Boundaries
The first octant is defined by the inequalities \(x \geq 0\), \(y \geq 0\), and \(z \geq 0\). Along with these, the cube is bounded by the planes at \(x=2\), \(y=2\), and \(z=2\). Thus, these introduce upper boundaries of \(x \leq 2\), \(y \leq 2\), and \(z \leq 2\).
3Step 3: Write the Inequalities
The inequalities that describe the cube in the problem are derived from the identified boundaries. The complete set of inequalities representing the cube are: \(0 \leq x \leq 2\), \(0 \leq y \leq 2\), and \(0 \leq z \leq 2\).
4Step 4: Verify the Solution
Check that the set of inequalities ensures the cube is in the first octant and bounded correctly. The inequalities reflect that the cube is between the origin and the planes \(x=2\), \(y=2\), and \(z=2\), ensuring it remains within the specified boundaries.
Key Concepts
InequalitiesFirst OctantThree-Dimensional Geometry
Inequalities
Inequalities are mathematical expressions that show the relationship between two values in terms of greater or less than. In three-dimensional coordinate geometry, they are often used to define regions or volumes in space.
For instance, when describing a solid shape like a cube, inequalities help to establish the limits in each dimension.
In the case of the cube in this problem:
Thus, inequalities concisely express spatial boundaries in mathematics.
For instance, when describing a solid shape like a cube, inequalities help to establish the limits in each dimension.
In the case of the cube in this problem:
- The inequalities ensure each coordinate (x, y, z) is non-negative, by indicating: \(x \geq 0\), \(y \geq 0\) and \(z \geq 0\).
- They also cap these coordinates, establishing that they cannot exceed 2, requiring: \(x \leq 2\), \(y \leq 2\), and \(z \leq 2\).
Thus, inequalities concisely express spatial boundaries in mathematics.
First Octant
The term "first octant" refers to one of the eight divisions in a three-dimensional coordinate system. The space is divided by the coordinate planes: the xy-plane, yz-plane, and xz-plane.
The first octant is where all three coordinates, x, y, and z, are non-negative. This is typical in first-year geometry problems because it's straightforward and limits values to positive ones, making calculations easier to manage.
In our problem, the cube exists entirely within the first octant, therefore the inequalities must reflect:
The first octant is where all three coordinates, x, y, and z, are non-negative. This is typical in first-year geometry problems because it's straightforward and limits values to positive ones, making calculations easier to manage.
In our problem, the cube exists entirely within the first octant, therefore the inequalities must reflect:
- \(x \geq 0\)
- \(y \geq 0\)
- \(z \geq 0\)
Three-Dimensional Geometry
Three-dimensional, or 3D, geometry deals with figures that have depth, in addition to length and width. Understanding shapes in three dimensions involves visualizing volumes and surfaces characterized by their lengths, widths, and heights.
A cube, one of the simplest 3D shapes, is defined by equal length sides. In this exercise, the cube is perfectly square between the bounds of the planes \(x = 0\) and \(x = 2\), \(y = 0\) and \(y = 2\), \(z = 0\) and \(z = 2\). This forms a neat 2-unit cube.
As geometry expands into three dimensions, the application of inequalities becomes crucial to capture the multitude of spaces and shapes that go beyond just a plane. The concept also extends to other shapes and structures, reinforcing the understanding of volume, area, and spatial reasoning. As you explore further, you'll find these basic principles of three-dimensional geometry are the foundation upon which much more complex geometry builds.
A cube, one of the simplest 3D shapes, is defined by equal length sides. In this exercise, the cube is perfectly square between the bounds of the planes \(x = 0\) and \(x = 2\), \(y = 0\) and \(y = 2\), \(z = 0\) and \(z = 2\). This forms a neat 2-unit cube.
As geometry expands into three dimensions, the application of inequalities becomes crucial to capture the multitude of spaces and shapes that go beyond just a plane. The concept also extends to other shapes and structures, reinforcing the understanding of volume, area, and spatial reasoning. As you explore further, you'll find these basic principles of three-dimensional geometry are the foundation upon which much more complex geometry builds.
Other exercises in this chapter
Problem 36
In Exercises \(33-38,\) find the distance from the point to the line. $$ (2,1,-1) ; \quad x=2 t, \quad y=1+2 t, \quad z=2 t $$
View solution Problem 36
Find a. the direction of \(\overrightarrow{P_{1} P}_{2}\) and b. the midpoint of line segment \(P_{1} P_{2}\). \(P_{1}(1,4,5) \quad P_{2}(4,-2,7)\)
View solution Problem 37
Find the areas of the parallelograms whose vertices are given in Exercises \(35-40 .\) $$ A(-1,2), \quad B(2,0), \quad C(7,1), \quad D(4,3) $$
View solution Problem 37
In Exercises \(33-38,\) find the distance from the point to the line. $$ (3,-1,4) ; \quad x=4-t, \quad y=3+2 t, \quad z=-5+3 t $$
View solution