Problem 12 - 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 \(-(1)\) . $$9 x^{2}+4 y^{2}+2 z^{2}=36$$

Step-by-Step Solution

Answer

The surface is an ellipsoid.

Step 1: Identify Standard Form

The given equation is \(9x^2 + 4y^2 + 2z^2 = 36\). To identify the surface type, we compare it to the standard form of an ellipsoid, which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).

Step 2: Simplify the Equation

Simplify the given equation by dividing all terms by 36, resulting in \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{18} = 1\).

Step 3: Interpret the Standard Form

Now, the equation \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{18} = 1\) is in the standard form for an ellipsoid. This confirms that the surface is an ellipsoid.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. There are four classic types: circles, ellipses, parabolas, and hyperbolas. When we're dealing with three-dimensional shapes, these basics extend into surfaces such as ellipsoids, paraboloids, and hyperboloids. Each type has a distinct equation form that can help us identify them.
In this exercise, the primary aim is to match the given equation with its proper surface type using these concepts. Understanding conic sections lays the groundwork for understanding more complex three-dimensional shapes that are often encountered in multivariable calculus.

• Circles are special types of ellipses where the two axes are equal, expressed by \(x^2 + y^2 = r^2\).
• Ellipses take the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) when centered at the origin.
• Parabolas are represented by equations like \(y = ax^2\) or \(x = ay^2\).
• Hyperbolas have forms such as \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Each form indicates how a plane slices through a cone, and who knew that understanding this concept could also help explore deeper into shapes like ellipsoids?

Surface Identification

Identifying a surface through its equation is an essential skill in geometry and calculus. This exercise provides us with the ellipsoid equation, recognizable by its standard form comparison.
Ellipsoids resemble stretched spheres, and their general equation is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\). To successfully identify a given equation as representing an ellipsoid, the task involves transforming or simplifying it into this recognizable standard form.

• Ellipsoid: All squared terms, with positive coefficients, and a constant on the other side of the equation. The denominators \(a^2, b^2, c^2\) determine the length of the semi-axes.
• Identify ellipsoid properties by comparing the simplified equation to key details within the general form.
• The symmetrical nature and bounded surface features stand out as unique traits of ellipsoids.

By analyzing the given equation, dividing it such that the right side becomes one, we can easily spot that it fits an ellipsoid pattern.

Equation Simplification

Simplifying equations to match a standard form is crucial for identifying 3D surfaces. In this exercise, we start with a more complex shape equation, \(9x^2 + 4y^2 + 2z^2 = 36\), and we simplify it to an ellipsoid form.
Here's how the simplification process works:

• Divide every term in the equation by the constant (36 here) to start reshaping it.
• This yields \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{18} = 1\), which is now comparable to our target form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).
• The terms under the fractions reflect the squared terms from an ellipsoid; thus, simplifying places the equation into a recognizable pattern.

Through these steps, students learn to simplify and identify commonly encountered geometric forms. Clever manipulation of equations can unveil the hidden types of surfaces they describe.