Problem 23 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) PARABOLOIDS AND CONES $$x=4-4 y^{2}-z^{2}$$

Step-by-Step Solution

Answer

The surface is an elliptic paraboloid opening along the negative \(x\)-axis with vertex at (4,0,0).

Step 1: Identify the Surface

The equation given is written in terms of three variables: \(x\), \(y\), and \(z\). The form \(x = 4 - 4y^2 - z^2\) is similar to that of an elliptic paraboloid. In general, similar surface equations are typically written as \(z = x^2/a^2 + y^2/b^2\) but have a flipped version because of the \(x\) term.

Step 2: Convert to Standard Form

To recognize the type of paraboloid, let's rearrange and rewrite the given equation as follows: \((4 - x) = 4y^2 + z^2\). This form helps to deduce that it's an elliptic paraboloid opening along the negative \(x\)-axis.

Step 3: Analyze the Shape Orientation

The fact that \(x\) decreases as \(y^2\) and \(z^2\) increase indicates that the paraboloid opens towards decreasing \(x\). The surface is centered around the line \(y = 0\), \(z = 0\) and its vertex is at \(x=4\), \(y=0\), \(z=0\).

Step 4: Sketch the Surface

Begin with the vertex at \((4, 0, 0)\) and sketch the paraboloid symmetrically around the \(x\)-axis. As \(x\) decreases, \(y^2\) and \(z^2\) must increase, indicating wider sections as you move away from \(x = 4\). The cross-sections parallel to the \(yz\)-plane (constant \(x\)) are ellipses that get smaller as \(x\) approaches 4.

Key Concepts

Surface Sketching

Surface sketching is a valuable tool in understanding three-dimensional shapes like elliptic paraboloids. An elliptic paraboloid is a surface that looks like an elongated bowl, with its cross-sections forming ellipses. In this exercise, the equation given is not in its usual form, so our first task is to interpret its structure. Start by identifying the vertex. Here, the vertex is located at the point \(4, 0, 0\). The surface is symmetric around the x-axis.
Elliptic paraboloids are symmetrical and have circular or elliptical cross-sections. Imagine taking slices parallel to the yz-plane (i.e., slicing at different x values while holding y and z constant). In this surface, as you move away from the vertex at \(x = 4\), these slices form ellipses. They become progressively larger as the value of x decreases, creating the bowl-like shape.

• Vertex: The starting point of the surface, located at \(x=4, y=0, z=0\).
• Opening Direction: Opens along the negative x-axis.
• Slices: As x decreases, ellipses grow larger.

Sketching this surface helps visualize the relationship between the equation and its geometric shape.

Equation Rearrangement

Rearranging equations is a crucial step in classifying surfaces and understanding their properties. The original equation is given as \(x = 4 - 4y^2 - z^2\). This might not directly resemble the standard form of an elliptic paraboloid equation, but rearranging can help.
By rewriting the equation to \(4 - x = 4y^2 + z^2\), we can interpret it more easily. Here, the transformation lets us see that it's an elliptic paraboloid opening along the negative x-axis. The term \(4 - x\) shows us how far along the axis the slice of the paraboloid is.
An understanding of rearrangement involves noting:

• The original equation’s structure compared to the standard form.
• How the arrangement clarifies the orientation and type of paraboloid.
• Recognizing that these adjustments align the surface with expected geometric features.

Rearranging is a powerful tool for mathematical visualization and understanding.

3D Shape Analysis

Analyzing 3D shapes like elliptic paraboloids helps us not only to visualize but also understand the behavior and properties of these surfaces. For \(x = 4 - 4y^2 - z^2\), we note that as y and z increase or decrease, they contribute to the reduction in the value of x.
The role of each variable is essential. \(y^2\) and \(z^2\) are contributing factors that automatically increase as y or z differs from zero, influencing the shape's width and orientation. The surface’s vertex is at \(4, 0, 0\), meaning when y and z are both zero, x is at its maximum value.
Elliptic paraboloids are characterized by:

• Symmetrical Nature: The symmetry allows similar behavior in responding to changes along all axes.
• Vertex: The peak point, from which the paraboloid opens outwards.
• Cross-Sections: Different cross-sections show ellipses along planes parallel to y and z axes.

Understanding these traits is vital in predicting how variations in equations translate to changes in the physical shape.