Problem 22 - Thomas Calculus

Question

Sketch the surfaces in Exercises \(13-44.\) PARABOLOIDS AND CONES $$z=8-x^{2}-y^{2}$$

Step-by-Step Solution

Answer

The surface is an elliptic paraboloid opening downward with a vertex at (0, 0, 8).

Step 1: Recognize the Type of Surface

The given equation is \( z = 8 - x^2 - y^2 \). This is in the standard form of an elliptic paraboloid given by \( z = c - ax^2 - by^2 \), where \( a, b > 0 \). This indicates a paraboloid that opens downwards.

Step 2: Identify Vertex and Axis

The vertex of the paraboloid is at the point where \( x = 0 \) and \( y = 0 \), which gives \( z = 8 \). So, the vertex is at \((0, 0, 8)\). The axis of symmetry is along the negative \( z \)-axis, as the paraboloid opens downward.

Step 3: Determine Intercepts

For the \( z \)-intercept, set \( x = 0 \) and \( y = 0 \) in the equation: \( z = 8 \). For \( x \)- and \( y \)-intercepts, set \( z = 0 \): \( 0 = 8 - x^2 - y^2 \) simplifying to \( x^2 + y^2 = 8 \). This is the equation of a circle in the \( xy \)-plane.

Step 4: Draw Level Curves

Level curves are obtained by setting \( z = k\), with \( 0 \leq k \leq 8 \). Substitute \( z = k \) into the equation to get \( k = 8 - x^2 - y^2 \), rearranging gives \( x^2 + y^2 = 8 - k \). These are circles with radii \( \sqrt{8-k} \) in the \( xy \)-plane.

Step 5: Sketch the Surface

Start by plotting the vertex at \( (0, 0, 8) \). Sketch circles of decreasing radius as \( z \) decreases from 8 to 0. Connect the circles smoothly to form the downward-opening paraboloid, ensuring the surfaces taper into a single point at each level curve.

Key Concepts

Vertex of a Paraboloid

In mathematics, the vertex of a paraboloid is a critical point that acts as the pinnacle or base of the shape, depending on its orientation. For an elliptic paraboloid given by the equation \( z = c - ax^2 - by^2 \), the vertex is located where the expressions for \( x \) and \( y \) are both zero. This simplifies the equation to \( z = c \), offering the vertex coordinates as \((0, 0, c)\). In this case, for the paraboloid described by \( z = 8 - x^2 - y^2 \), the vertex is at \((0, 0, 8)\).

• The vertex is crucial because it provides a reference point from which we determine other aspects of the paraboloid, such as its spread and direction.
• Understanding the vertex can help in graphing and analyzing how the surface behaves as \( x \) and \( y \) change.

Axis of Symmetry

The axis of symmetry for a paraboloid is the line around which the surface is symmetrical. For elliptic paraboloids of the form \( z = c - ax^2 - by^2 \), the axis of symmetry aligns with the \( z \)-axis. This particular paraboloid opens downward along the negative \( z \)-axis since the coefficients of \( x^2 \) and \( y^2 \) are both positive, indicating that it curves away from the vertex.

• The axis of symmetry helps in visualizing the shape since the paraboloid is uniform around this axis.
• By knowing the axis, students can better understand the slope and spread of the paraboloid.

The axis of symmetry is vital for creating a clear and precise graph of the paraboloid surface.

Level Curves

Level curves are two-dimensional slices of a three-dimensional surface that help us understand the structure of a paraboloid. By setting \( z = k \) for various values of \( k \), we can find sets of \( (x, y) \) that satisfy \( x^2 + y^2 = 8 - k \). These equations describe circles in the \( xy \)-plane with radii \( \sqrt{8-k} \).

• As \( k \) decreases from 8 to 0, the radii of these level curves shrink, showing how the paraboloid narrows as it gets closer to the vertex.
• Level curves are instrumental in three-dimensional graphing, offering a step-by-step path to constructing the full surface from its slices.

By examining level curves, we can grasp both the height and variation of the surface, making it a powerful tool for visualization.

Three-Dimensional Graphing

Three-dimensional graphing involves visualizing surfaces like paraboloids in a coordinate system made of three axes: \( x \)-axis, \( y \)-axis, and \( z \)-axis. For the paraboloid \( z = 8 - x^2 - y^2 \), the graphing process starts with plotting the vertex at \( (0, 0, 8) \). Each level curve, which you obtain by setting different \( z \) values, can be plotted as circles in the \( xy \)-plane. These circles connect to form a downward-opening surface.

• Visualizing through three-dimensional graphing offers insights into the symmetry and shape of the paraboloid.
• Using this approach, you can accurately depict both the shape's curvature and its diminishing size as it moves toward the vertex.

Three-dimensional graphing not only aids in understanding the mathematics behind the paraboloid but also enhances intuitive comprehension by providing a tangible view of the surface.