Problem 23 In Exercises $17-24,$ describe... [FREE SOLUTION]

Chapter 12: Problem 23

In Exercises $17-24,$ describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \text { a. } y \geq x^{2}, \quad z \geq 0 \quad \text { b. } x \leq y^{2}, \quad 0 \leq z \leq 2 $$

Short Answer

Expert verified

a. Region above parabola y = x虏, z 鈮� 0; b. Region left of paraboloid x = y虏, 0 鈮� z 鈮� 2.

Step by step solution

Interpret the Inequality for Part a

The inequality $ y \geq x^2 $ represents the space above the parabola $ y = x^2 $ in the xy-plane. This set includes all points $(x, y)$ where the value of $ y $ is greater than or equal to $ x^2 $.

Interpret the Inequality for z in Part a

The inequality $ z \geq 0 $ implies that all points have a zero or positive z-coordinate. This condition defines the region as including the entire half-space above and including the plane z = 0 (the xy-plane).

Combine Conditions for Part a

By combining $ y \geq x^2 $ and $ z \geq 0 $, the set of points is a semi-infinite region that is above the parabola $ y = x^2 $ and at or above the xy-plane, extending infinitely upward in the positive z-direction.

Interpret the Inequality for Part b

The inequality $ x \leq y^2 $ represents the space to the left of the paraboloid $ x = y^2 $ in 3D space. This includes all points $(x, y, z)$ where the value of $ x $ is less than or equal to $ y^2 $.

Interpret the Conditions for z in Part b

The condition $ 0 \leq z \leq 2 $ confines the z-coordinate of the points between 0 and 2, inclusive. Thus, points can lie anywhere between these two planes: the xy-plane (z=0) and the plane z=2.

Combine Conditions for Part b

The solution is the set of all points $(x, y, z)$ that satisfy $ x \leq y^2 $ and have a z-coordinate that lies between 0 and 2, inclusive. This space is bounded vertically between z=0 and z=2 and extends infinitely to the left of $ x = y^2 $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequalities in 3D

Inequalities in three-dimensional space define regions where certain conditions are met. These inequalities extend concepts from 2D to 3D by involving all three coordinates: x, y, and z. For example:

The condition $ y \geq x^2 $ describes a region above the curve $ y = x^2 $ in the xy-plane. This means that y can be equal to or greater than the square of x, giving a region that looks like a bowl opening upward.
The inequality $ z \geq 0 $ indicates a half-space above the xy-plane. Here, z must be zero or positive, meaning the region includes and rises from the xy-plane upwards into all positive values of z.
For instance, when we consider both $ x \leq y^2 $ and $ 0 \leq z \leq 2 $, we find a region extending to the left of the paraboloid $ x = y^2 $, but capped between the planes z = 0 and z = 2.

Combining these inequalities helps to describe distinct regions of space within defined boundaries, revealing how multiple planes and surfaces interact with one another.

Paraboloid Regions

Paraboloid regions arise in 3D space when one coordinate, typically x or y, is related to the square of another coordinate. Such relationships create shapes called paraboloids, an extension of the 2D parabola into the third dimension:

Imagine the surface described by $ x = y^2 $; this geometric shape is a paraboloid. Every point on this surface has coordinates where x equals the square of y.
A region defined by $ x \leq y^2 $ is the space left of and including this paraboloid in 3D space, indicating a volumetric area that curves outward as it extends along the y-axis.

Picturesquely, it's like a hot air balloon lying on its side. Part of understanding paraboloid regions involves visualizing how they pivot and stretch within the three axes of the coordinate system.

Half-Space Concept

The half-space concept in mathematics refers to dividing 3D space into two distinct regions by using a plane:

The equation of a plane, such as $ z = 0 $, acts as a boundary that separates space into two half-spaces. One side will satisfy $ z \geq 0 $, while the opposite satisfies $ z < 0 $.
In this case, $ z \geq 0 $ represents the half-space above the xy-plane, including the plane itself. This means that any point's z-coordinate must be non-negative.
Additional conditions, like $ 0 \leq z \leq 2 $, further restrict this half-space into a specific slice. Here, the half-space is restricted vertically between two flat planes at z = 0 and z = 2.

The half-space concept is crucial in understanding and solving problems involving spatial inequalities, as it provides a fundamental way to describe and visualize regions above, below, or between given planes.

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91影视

In Exercises \(17-24,\) describe the sets of points in space whose coordinates satisfy the given inequalities or combinations of equations and inequalities. $$ \text { a. } y \geq x^{2}, \quad z \geq 0 \quad \text { b. } x \leq y^{2}, \quad 0 \leq z \leq 2 $$

Short Answer

Step by step solution

Interpret the Inequality for Part a

Interpret the Inequality for z in Part a

Combine Conditions for Part a

Interpret the Inequality for Part b

Interpret the Conditions for z in Part b

Combine Conditions for Part b

Key Concepts

Inequalities in 3D

Paraboloid Regions

Half-Space Concept

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