Problem 61 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-76\) $$ 9 x^{2}+4 y^{2}+z^{2}=36 $$

Step-by-Step Solution

Answer

It's an ellipsoid centered at the origin, stretched more along the z-axis.

Step 1: Identify the Surface Type

The given equation is \(9x^2 + 4y^2 + z^2 = 36\). This represents a quadratic surface. By dividing through by 36 to compare with the standard form of an ellipsoid, we get \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{36} = 1\). This equation is the standard form of an ellipsoid.

Step 2: Identify the Axes

From the equation \(\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{36} = 1\), identify the semi-axis lengths. The coefficients \(\frac{1}{4}\), \(\frac{1}{9}\), and \(\frac{1}{36}\) correspond to the terms \(x^2\), \(y^2\), and \(z^2\) respectively, suggesting a semi-major axis along the z-axis and semi-minor axes along the x and y axes. The semi-axis lengths are \(2\) along x, \(3\) along y, and \(6\) along z.

Step 3: Sketch the Surface

Sketch an ellipsoid centered at the origin (0, 0, 0) with axes of length 4 along the x-direction, 6 along the y-direction, and 12 along the z-direction (since these are the full lengths of each axis, double the semi-axis lengths). The ellipsoid is symmetric about the origin, so ensure that each semi-axis is drawn to scale. The surface will look like an elongated sphere, stretched most along the z-axis.

Key Concepts

Quadratic surfaces

Quadratic surfaces are an essential concept in three-dimensional geometry and are formed by equations involving squares of variables. These surfaces are an extension of conic sections into three dimensions. Here are some common types of quadratic surfaces:

• Ellipsoids
• Hyperboloids
• Paraboloids

Quadratic surfaces are characterized by equations that often include squared terms, like in our example, which looks like: \[ Ax^2 + By^2 + Cz^2 + ext{other terms} = 0 \] In our exercise, the equation given is: \[ 9x^2 + 4y^2 + z^2 = 36 \] To classify this as an ellipsoid, we reformat it into its standard form. Recognizing these basic surface types helps with visualizing and solving geometric problems in multi-dimensional space.

Standard form

The standard form of a quadratic surface is critical for understanding its type and dimensions. The standard form for an ellipsoid is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \] Here, \(a\), \(b\), and \(c\) represent the lengths of the semi-axes. Transforming our exercise equation to this form involves dividing all terms by 36, yielding:\[ \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{36} = 1 \]This equation now clearly demonstrates the standard form of an ellipsoid. Recognizing standard forms allows for easier comparison and identification, aiding in both solving and sketching. It simplifies the process of determining dimensions and orientation of the surface.

Axes of ellipsoid

Once an equation is in standard form, identifying the axes of the ellipsoid becomes straightforward. In our case, the equation is: \[ \frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{36} = 1 \]The denominators \(4\), \(9\), and \(36\) provide us with the semi-axis lengths: 2, 3, and 6, respectively. Here's how you interpret these values:

• \(2\) represents the semi-axis length along the x-axis.
• \(3\) represents the semi-axis length along the y-axis.
• \(6\) represents the semi-axis length along the z-axis, which is the longest, indicating the major axis.

Identifying these axes is crucial for understanding the ellipsoid's orientation and dimensions, guiding the sketching process effectively.

Surface sketching

Sketching an ellipsoid requires careful attention to its axes. Begin with the centered position at the origin \((0, 0, 0)\). From there, extend lines along each axis using the full length of the axes, which are twice the semi-axis lengths:

• The x-axis extends to \(-2, 2\).
• The y-axis extends to \(-3, 3\).
• The z-axis, being the longest, extends to \(-6, 6\).

This creates an elongated sphere, with symmetry about the origin, and the longest stretch occurs along the z-axis. When sketching, maintaining the proportions of these axes is key, as it ensures the ellipsoid's shape is accurately represented. Visualizing this surface effectively aids in comprehending its spatial properties.