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

Ellipsoid

Step 1: Identify the General Form

The given equation is \(9x^2 + 4y^2 + 2z^2 = 36\). This equation is a quadratic form, which suggests that it may represent a type of ellipsoid if we can bring it into a standard form.

Step 2: Bring Equation to Standard Form

To bring the equation to a standard form, divide all terms by 36: \[ \frac{9x^2}{36} + \frac{4y^2}{36} + \frac{2z^2}{36} = 1\]. Simplifying, we get \[ \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{18} = 1 \].

Step 3: Identify the Surface Type

The equation \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{18} = 1\) is in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\), which characterizes an ellipsoid because all the coefficients are positive, and the sum equals 1.

Step 4: Conclusion

Thus, the surface defined by the equation \(9x^2 + 4y^2 + 2z^2 = 36\) is an ellipsoid.

Key Concepts

Ellipsoid

An ellipsoid is a three-dimensional surface, which can be thought of as a stretched or compressed sphere. It is symmetrical along its three principal axes. Imagine you have a ball made of clay, and you slightly press from one side, causing it to bulge on another side. That’s a simple way to think of an ellipsoid.
An ellipsoid can come in different shapes depending on the length of its principal axes. There are three main types you might encounter:

• Prolate Ellipsoid - Looks like an elongated sphere; the major axis is longer than the other two axes.
• Oblate Ellipsoid - Slightly flattened, where one axis is shorter, making the equatorial radius larger than the polar radius.
• Spheroid - When all three axes are equal, it is not elongated or flattened, representing a perfect sphere.

A key feature of an ellipsoid is that its points are equidistant from the center, considering its radii along each axis. This makes them very useful in fields like physics and astronomy where shapes of celestial bodies are calculated.

Equation Standard Form

The equation of an ellipsoid in its standard form is quite specific. By standardizing, it becomes easier to identify and compare.
The general form of the equation is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \]Here:

• \(a\), \(b\), and \(c\) represent the semi-axes lengths of the ellipsoid along the x, y, and z axes, respectively.
• It’s important that all fractions add up to 1.
• Each term represents a specific dimension of the ellipsoid.

By converting any given quadratic equation into this form, we can easily determine it is an ellipsoid. This conversion often involves dividing each term by the constant on the right-hand side, so the equation equals 1.

Surface Type Identification

Identifying the surface type from a quadratic equation involves analyzing the coefficients and terms. Understanding the pattern can help you decipher what 3D shape the equation represents.
For an ellipsoid, you'll notice a few characteristics:

• All quadratic terms \(x^2\), \(y^2\), and \(z^2\) are positive, indicating that these dimensions are real and positive.
• There is no linear term present, such as \(x\), \(y\), or \(z\), which means the ellipsoid is centered at the origin without any shift.
• The constants squared in the denominators reveal the lengths of the axes, which help in determining the type of ellipsoid (prolate, oblate, spheroid).

By recognizing these clues, you can confidently identify the 3D surface type from any quadratic equation. This ability is very valuable in calculus and analytical geometry fields.