Question

Think About It In Exercises \(65-68\), find inequalities that describe the solid, and state the coordinate system used. Position the solid on the coordinate system such that the inequalities are as simple as possible. The solid between the spheres \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=9,\) and inside the cone \(z^{2}=x^{2}+y^{2}\)

Short answer

The inequalities that describe the solid are \(4 \leq x^{2}+y^{2}+z^{2} \leq 9\) and \(z \geq \sqrt{x^{2}+y^{2}}\). The used coordinate system is three-dimensional Cartesian coordinate system. The solid is positioned on the origin (0,0,0)

Step-by-step solution

Identify the inequalities

The equations for the two spheres are \(x^{2}+y^{2}+z^{2}=4\) and \(x^{2}+y^{2}+z^{2}=9\). Therefore, the inequalities that describe the solid between these two spheres are \(4 \leq x^{2}+y^{2}+z^{2} \leq 9\). The equation for the cone is \(z^{2}=x^{2}+y^{2}\) or in other words, \(z=\sqrt{x^{2}+y^{2}}\). Since the solid is inside the cone, the inequality representing this constraint is \(z \geq \sqrt{x^{2}+y^{2}}\)

State the coordinate system used

The coordinate system being used is the three-dimensional Cartesian coordinate system which uses coordinates (x, y, z).

Position the solid on the coordinate system

The solid is positioned on the coordinate system such that the origin (0,0,0) is at the center of both spheres and the tip of the cone. This positioning makes the inequalities as simple as possible as the terms in the equations and inequalities do not have added or subtracted constants.

Key concepts

Cartesian Coordinate System

The Cartesian coordinate system is a foundational element in geometry and calculus. It consists of two or three perpendicular axes, commonly referred to as x, y, and z. In a three-dimensional space, the x, y, and z axes allow for the precise positioning of points and the representation of shapes and solids.

When dealing with solid geometry inequalities, placing the solid of interest, such as the one bounded between two spheres in the exercise, at the origin can simplify the representation of this solid. By leveraging the symmetrical properties of the sphere and the cone, the Cartesian coordinate system allows the equations to be described without added or subtracted constants, resulting in cleaner and more straightforward inequalities.

Spherical Coordinates

While the Cartesian coordinate system describes points in space using perpendicular axes, spherical coordinates offer a different approach by using a system of radial and angular coordinates. It essentially breaks down a point's location into its radial distance from the origin, the angle with the positive z-axis (polar angle), and the angle with the positive x-axis in the xy-plane (azimuthal angle).

For the given solid between two spheres, transitioning into spherical coordinates could prove useful, particularly for integrating over the volume. The presence of spherical symmetry in the inequalities from the exercise would make spherical coordinates an elegant and efficient way to describe the region. However, when looking to express simple inequalities, particularly in the context of the exercise where the central idea is to find inequalities in 'as simple as possible' form, the Cartesian coordinate system is preferred.

Inequality Representation in Calculus

In calculus, inequalities are used to represent the bounds of a particular region or solid. They define the conditions that the points inside the region must satisfy. As seen in the solution to the textbook exercise, the inequalities for the solid are given by two conditions: one that describes it as being outside the smaller sphere and inside the larger sphere, and another that describes it as being within a cone.

The exercise simplifies the description of the solid through the use of inequalities without additional constants, which is a strategy often applied in calculus to minimize computation complexity. This simplification is vital for tasks like integration, where capturing the essence of the solid without unnecessary complications can make the difference between a solvable problem and a cumbersome task.