Problem 92 - Thomas Calculus

Question

Use a CAS to plot the surfaces in Exercises \(89-94\) . Identify the type of quadric surface from your graph. $$ \frac{y^{2}}{16}=1-\frac{x^{2}}{9}+z $$

Step-by-Step Solution

Answer

The surface is an elliptic paraboloid.

Step 1: Rewrite the Equation

First, we'll rewrite the given equation in a more standard form to make it easier to identify the type of quadric surface. The original equation is \( \frac{y^2}{16} = 1 - \frac{x^2}{9} + z \). Let's isolate \( z \) on one side:\[ z = \frac{y^2}{16} - 1 + \frac{x^2}{9} \]This rewritten equation is more suitable for plotting in a CAS.

Step 2: Recognize the Quadric Surface Type

Now that we have \( z = \frac{y^2}{16} - 1 + \frac{x^2}{9} \), compare this with the standard form of quadric surfaces: \( z = ax^2 + by^2 + cz \). This equation represents a paraboloid because it involves both \( x^2 \) and \( y^2 \) terms with positive coefficients.

Step 3: Use a CAS to Plot the Surface

Open a Computer Algebra System (CAS) such as GeoGebra, Mathematica, or Desmos. Input the equation \( z = \frac{y^2}{16} - 1 + \frac{x^2}{9} \) and generate the plot. You should see a paraboloid shape, opening along the \( z \)-axis.

Step 4: Analyze the Graph

Observe the 3D plot generated by the CAS. The surface should appear as a paraboloid, specifically an elliptic paraboloid, due to the elliptical cross-section in the \( xy \)-plane.

Key Concepts

Elliptic Paraboloid

An elliptic paraboloid is a type of quadric surface that resembles the shape of a typical satellite dish or a bowl. The general equation for an elliptic paraboloid is given by \( z = ax^2 + by^2 \), where both \( a \) and \( b \) are positive constants. In the exercise, the equation \( z = \frac{y^2}{16} - 1 + \frac{x^2}{9} \) represents an elliptic paraboloid because it follows this form.

Key features of an elliptic paraboloid include:

• It has a single vertex, which acts as the point of symmetry.
• The cross-sections parallel to the \( xy \)-plane are ellipses, leading to its name.
• The elliptic paraboloid can either open upwards or downwards, depending on the signs of the coefficients \( a \) and \( b \).

Understanding these characteristics helps in identifying elliptic paraboloids easily in mathematical exercises and real-world applications.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a powerful tool used to perform complex algebraic operations digitally. CAS software can manipulate symbolic equations, thus providing exact solutions and enabling visualization of 3D models.

Commonly used CAS tools include GeoGebra, Mathematica, and Desmos. These platforms allow users to input mathematical expressions and generate graphical outputs instantly. In this exercise, using a CAS to plot the surface of the equation assists in identifying the type of quadric surface as an elliptic paraboloid.

Benefits of using a CAS:

• It simplifies complex algebraic calculations.
• It provides visual representation which enhances understanding.
• It saves time and reduces potential errors in manual calculations.

Mastery of a CAS can significantly aid students in solving mathematical problems efficiently.

3D Plotting

3D plotting is an essential skill for visualizing functions and surfaces in three-dimensional space. It provides an interactive way to explore mathematical concepts by allowing one to view and manipulate the graphical output.

In the context of the exercise, 3D plotting helps to appreciate the structure and form of the elliptic paraboloid. The equation \( z = \frac{y^2}{16} - 1 + \frac{x^2}{9} \) can produce a 3D model of the surface when entered into a CAS.

Important aspects of 3D plotting include:

• Determining the ranges for the axes to capture the full scope of the graph.
• Choosing the appropriate plot resolution for clarity.
• Using rotation and zoom features to examine different perspectives.

Such visual analysis is crucial in connecting theoretical math with tangible understanding.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. They consist of various shapes like ellipses, parabolas, circles, and hyperbolas, each with unique properties. Conic sections form an integral part of studying quadric surfaces.

In the context of an elliptic paraboloid, the cross-sections parallel to the \( xy \)-plane are ellipses, thereby linking it to conic sections. This property is pivotal in characterizing the elliptic paraboloid among other quadric surfaces.

Conic sections are important because:

• They appear in numerous scientific and engineering applications, from satellite dishes to architectural designs.
• They provide foundational concepts in topics like projective geometry and optics.
• They enable deeper insight into the nature of quadratic equations and their geometric interpretations.

Understanding conic sections allows students to grasp the geometric aspects of algebraic equations effectively.