Problem 4 - Thomas Calculus

Question

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ y^{2}+z^{2}=x^{2} $$

Step-by-Step Solution

Answer

The equation \(y^2 + z^2 = x^2\) represents a cone.

Step 1 - Identify the Standard Form

The given equation is \(y^2 + z^2 = x^2\). This is a standard form often related to conic sections, specifically, cones and cylinders.

Step 2 - Simplify the Equation

Rewrite the equation \(y^2 + z^2 = x^2\) in a form that is easier to compare with standard types. Notice both sides are squared quantities which implies a relationship hinting at conical type surfaces.

Step 3 - Recognize the Surface Type

Compare \(y^2 + z^2 = x^2\) with the equation of a cone, which in standard form is \(Ax^2 + By^2 = Cz^2\) or similar permutations. This equation indeed represents a cone because it describes a relation where the sum of squares of two variables is equal to the square of the third.

Key Concepts

Equations of Surfaces

Equations of surfaces represent how various kinds of shapes can be described in three-dimensional space using mathematical formulas.
These shapes, such as spheres, planes, cylinders, and cones, have specific standard forms that let us identify them easily.
Surfaces are often described by equations involving squares of variables, as these match the geometric properties of 3D shapes.

• Quadratic Form: Most 3D surfaces come from quadratic equations. Quadratic means involving terms like square of a variable, such as \( x^2 \) or \( y^2 \).
• Symmetry: Equations of symmetric surfaces will look similar across axes, for example, spheres and cones.
• Visualization: To understand these surfaces, imagine graphing them in a 3D coordinate space.

For instance, the equation \( y^2 + z^2 = x^2 \) describes a unique shape in space, which we will explore further in relation to cones.

Cone

A cone is a 3D geometric shape with a circular base and a pointed top called the apex. Imagine a real-life cone, like an ice cream cone.
Mathematically, a cone in 3D space can be represented by an equation where the square of two variables adds up to the square of another.
The equation \( y^2 + z^2 = x^2 \) is a classic example of a cone's equation.

• Structure: A cone opens in both directions along the axis of the variable not squared. Here, that axis is the \( x \)-axis.
• Symmetry: Cones are symmetrical around their primary axis, which simplifies identifying them from their equations.
• Real-world examples: Think of a funnel or the type of cone used in traffic, which similarly widen from a point.

Understanding cones using these standard forms allows us to visualize them as endless 3D shapes, specified by one central axis and symmetric parts along other axes.

Standard Form of Equations

The standard form of an equation is a way of writing equations so they are easily recognizable and comparable.
For conic sections in 3D, this typically involves terms that are squared and provide geometric insight.
Let's explore the importance of writing these equations in standard form using the equation from our exercise: \( y^2 + z^2 = x^2 \).

• Generalization: In mathematics, standard forms help create a "template" that makes recognizing patterns easier. They're pivotal for classification.
• Comparison: These forms enable simple comparison with other equations to determine the type (e.g., cone, cylinder).
• Application: In geometry, writing equations in standard form aids in visualizing and solving real-world problems.

The equation \( y^2 + z^2 = x^2 \) is readily recognizable as a cone's equation because it matches the standard template where the sum of the squares of two variables equals the square of another. Understanding and utilizing the standard form helps streamline solving and identifying problems in geometry.