Question

Describe geometrically the level surfaces for the functions. \(f(x, y, z)=100 x^{2}+16 y^{2}+25 z^{2} ; k>0\)

Short answer

The level surfaces are ellipsoids centered at the origin.

Step-by-step solution

Identify the Type of Function

The given function is of the form \(f(x, y, z) = 100x^2 + 16y^2 + 25z^2\). This is a quadratic function in three variables, representing a quadratic surface.

Set the Function Equal to k

To describe the level surfaces, set the function equal to a constant \(k\), which is \(k = 100x^2 + 16y^2 + 25z^2\).

Reorganize in Terms of a Standard Form

Factor out the coefficients to express the equation in the form of an ellipsoid: \(\frac{x^2}{\frac{k}{100}} + \frac{y^2}{\frac{k}{16}} + \frac{z^2}{\frac{k}{25}} = 1\).

Interpret the Standard Form

The equation \(\frac{x^2}{\frac{k}{100}} + \frac{y^2}{\frac{k}{16}} + \frac{z^2}{\frac{k}{25}} = 1\) represents an ellipsoid centered at the origin. The denominators \(\frac{k}{100}, \frac{k}{16}, \frac{k}{25}\) are the squares of the semi-axes lengths along the x, y, and z-axes, respectively.

Key concepts

Quadratic Functions

Quadratic functions are mathematical expressions where the variables are raised to the second power, which means the highest degree of any variable is two. These functions are very versatile and can describe a variety of geometrical shapes and surfaces.

In the exercise, the function given is \(f(x, y, z) = 100x^2 + 16y^2 + 25z^2\). This is an example of a quadratic function in three variables: \(x\), \(y\), and \(z\).

Quadratic functions are not just about squares; they can also represent surfaces and curves, such as parabolas, ellipses, and hyperbolas. By setting the quadratic expression equal to a constant value, we can derive these shapes as level surfaces.

Ellipsoid

An ellipsoid is a three-dimensional geometric figure, similar to a sphere but stretched or compressed along its axes. It looks like an oval or an elongated ball.

In the context of the problem, the function can be rewritten to \[ \frac{x^2}{\frac{k}{100}} + \frac{y^2}{\frac{k}{16}} + \frac{z^2}{\frac{k}{25}} = 1 \] and this equation represents an ellipsoid.

For an ellipsoid aligned along the coordinate axes, each axis has a length determined by dividing \(k\) by the coefficients of \(x^2\), \(y^2\), and \(z^2\). Thus, this ellipsoid is centered at the origin (0,0,0) in three-dimensional space, with specific lengths for each axis depending on the constant \(k\).

Quadratic Surface

A quadratic surface is a surface that can be defined by a quadratic equation in three variables. These surfaces can take various forms, such as ellipsoids, paraboloids, and hyperboloids, which are dependent on the coefficients of the equation.

In the exercise, the function \(100x^2 + 16y^2 + 25z^2 = k\) describes a type of quadratic surface known as an ellipsoid.

Quadratic surfaces are integral in fields like geometry, physics, and engineering, as they help model and simulate real-world phenomena. For instance, ellipsoids can represent planets, atoms, or even light beams. Understanding how to visualize these surfaces is crucial for applications such as computer graphics and structural engineering.

Mathematical Visualization

Mathematical visualization is the process of creating a visual image or model to understand and explore mathematical concepts. It is a powerful tool for interpreting complex functions and surfaces.

By employing mathematical visualization, one can better comprehend the nature of the ellipsoid derived in the exercise. Visualizing the ellipsoid means seeing it as a three-dimensional shape centered at the origin, with axes stretched to different lengths based on the constants in the equation.

• Visualization helps in grasping how changing values of \(k\) affect the size and proportions of the ellipsoid.
• It also assists in appreciating how the alignment with axes changes with different configurations.

Mathematical visualization aids in making abstract mathematical ideas more tangible, thereby fostering deeper comprehension and learning.