Problem 25 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) PARABOLOIDS AND CONES $$x^{2}+y^{2}=z^{2}$$

Step-by-Step Solution

Answer

The surface is a right circular cone centered at the origin, extending along the z-axis.

Step 1: Identify the type of surface

The given equation is \(x^2 + y^2 = z^2\), which resembles the equation of a cone. This is a right circular cone centered along the z-axis because both \(x^2\) and \(y^2\) terms are present, and \(z\) is also squared.

Step 2: Analyze the coefficients and constraints

Note that the coefficients of \(x^2\), \(y^2\), and \(z^2\) are all equal, implying a cone with a symmetrical opening. Since there are no additional constants, the cone's vertex is at the origin \((0,0,0)\).

Step 3: Determine the orientation and opening

This cone opens symmetrically in the positive and negative z-directions because squaring \(z\) implies both positive and negative values for \(z\). As \(z\) increases, both \(x\) and \(y\) can take on larger values while maintaining equality in the equation, indicating a widening cone upwards and downwards.

Step 4: Sketch the cone

Draw a symmetrical cone with its vertex at the origin. The cone should extend equally along the positive and negative z-axis. The cross-sections parallel to the base (perpendicular to the z-axis) are circles whose radius increases with \(|z|\).

Key Concepts

Right Circular Cone

A right circular cone is a three-dimensional geometric shape that is formed by connecting a circle to a point, called the apex, which is not in the same plane as the circle. In our exercise, the mathematical equation given is \(x^2 + y^2 = z^2\). Here, this equation represents a right circular cone extending along the z-axis.- The cone is termed *right* because the apex is directly above the center of the base circle, making the axis of the cone perpendicular to the base.- The term *circular* comes from the base being a circle. This is what gives it the *right circular cone* name.The equation \(x^2 + y^2 = z^2\) tells us that for each coordinate \((x, y)\), there exists a \(z\) value satisfying the provided conditions, forming the typical bell shape of a cone with the vertex at the origin \((0,0,0)\). It is important to recognize that the cone will extend symmetrically on both sides of the \(xy\)-plane, creating a double-napped cone.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This mathematical field is vital as it allows us to describe geometric shapes algebraically.- In the case of our right circular cone \(x^2 + y^2 = z^2\), we are using a three-dimensional Cartesian coordinate system.- The variables \(x\), \(y\), and \(z\) each represent one of the axes in three-dimensional space, bringing to life complex shapes.With coordinate geometry, we can analyze the positioning, orientation, and dimension of the cone mathematically. This helps in understanding how the cone behaves in space and how it relates to other geometric entities. By plotting points that satisfy the cone equation on the coordinate grid, one can visualize how the cone stretches along the z-axis in 3D space.

Surface Sketching

Surface sketching is an essential skill when working with equations involving three variables. It involves visualizing and drawing the representation of geometric shapes, like cones, in three-dimensional space.- For the equation \(x^2 + y^2 = z^2\), surface sketching requires plotting sections of the cone to show its overall form.- Starting at the origin, since \((0,0,0)\) is the vertex, draw lines extending outward along the z-axis. The cone will expand as \(|z|\) increases.When sketching, it's helpful to consider cross-sections. In this case, sections perpendicular to the z-axis form circles, helping you see the cone's widening nature. Surface sketching involves imagining the intersection of planes to form slices of the cone that are easy to represent on paper, making the abstract concept of a three-dimensional cone tangible.