Problem 25 - University Calculus: Early Transcendentals

Question

Sketch the surfaces PARABOLOIDS AND CONES $$x^{2}+y^{2}=z^{2}$$

Step-by-Step Solution

Answer

The surface is a bi-directional circular cone centered along the z-axis, symmetric around the origin.

Step 1: Identify the Surface Type

The given equation is a 3D surface equation usually associated with a cone. It resembles the equation of a cone in the form \(x^2 + y^2 = z^2\). This type indicates a circular cone centered along the z-axis in a 3-dimensional space.

Step 2: Understand the Symmetry

Recognize that the equation \(x^2 + y^2 = z^2\) implies rotational symmetry around the z-axis. This means the shape will be consistent around this axis, forming a circular shape at each level of z.

Step 3: Consider the Cross Sections

For each constant value of \(z\), the cross-section of the surface is a circle centered at the origin in the xy-plane, with radius \(z\). This is derived by rewriting the equation as \(x^2 + y^2 = z^2\), indicating for every \(z\), \(x^2 + y^2 = r^2\).

Step 4: Sketch the Surface

Start by drawing the z-axis as the vertical axis and plot the circles corresponding to different levels of \(z\). At \(z=0\), the circle reduces to just a point at the origin. As \(z\) increases, larger circles are formed, creating the sides of the cone. This pattern extends infinitely both upward and downward, making the cone bi-directional.

Step 5: Highlight Cone Features

Emphasize that the vertex of the cone is at the origin \((0, 0, 0)\) and the cone opens symmetrically upwards and downwards along the z-axis. This completeness around the z-axis and symmetry around the origin defines the double cone.

Key Concepts

Conic Sections

Conic sections are shapes we get when we intersect a plane with a cone. These include circles, ellipses, parabolas, and hyperbolas. Each type is determined by the angle at which the plane cuts through the cone.

• If the plane cuts parallel to the base of the cone, it forms a circle.
• An angle more vertical than the circle but less than the angle of the side of the cone gives an ellipse.
• A plane parallel to one side of the cone produces a parabola, forming a U-shaped curve.
• When the plane cuts through both nappes (or halves) of a double cone, the shape is a hyperbola.

These sections have direct applications in engineering and physics, helping model dish antennas, car headlights, and orbits of celestial objects. For a cone defined by the equation \(x^2 + y^2 = z^2\), understanding these conic sections can enhance comprehension of different 3D slices through the cone.

Rotational Symmetry

Rotational symmetry occurs when a shape looks the same after a certain amount of rotation. Imagine spinning an object around an axis, and noticing that the shape doesn't change, staying identical.
In the equation \(x^2 + y^2 = z^2\), the rotational symmetry is around the z-axis. As you rotate the cone along this axis, the shape remains consistent at every angle because each cross-section is a circle centered at the origin.

• Each circle has the same radius equal to the absolute value of \(z\).
• This symmetry gives the cone a uniform appearance no matter how it is turned around the z-axis.

Understanding rotational symmetry is crucial in fields involving motion and balance, such as designing wheels or turbines, where identical distribution around an axis is essential.

Cone Geometry

Cone geometry revolves around understanding the structure and properties of cones. A cone has a circular base and a tip called a vertex. The geometry includes its volume, surface area, and other properties.
For the equation \(x^2 + y^2 = z^2\):

• The vertex is at the origin \((0, 0, 0)\), marking the top point of the cone.
• The cone opens along the positive and negative z-axis, making it bi-directional.
• The radius of each circular cross-section is \(z\), showing how the size changes with \(z\).
• The height between these sections varies as we move along the z-axis.

Cones are important in real-world construction and design, like traffic cones or conical flasks, helping us understand space, volume, and material efficiency. Familiarizing with cone geometry enhances spatial visualization.