Problem 43 - University Calculus: Early Transcendentals

Question

Sketch the surfaces ASSORTED $$4 y^{2}+z^{2}-4 x^{2}=4$$

Step-by-Step Solution

Answer

The equation represents a hyperboloid of one sheet.

Step 1: Identifying the Type of Surface

The given equation is \(4y^2 + z^2 - 4x^2 = 4\). We can identify the type of surface by rewriting it in a form that resembles known conic sections. Notice the signature of the equation suggests an elliptic hyperboloid due to the one negative squared term \(-4x^2\) and two positive squared terms \(4y^2 + z^2\).

Step 2: Normalizing the Equation

Divide the entire equation by 4 to simplify: \[ \frac{4y^2}{4} + \frac{z^2}{4} - \frac{4x^2}{4} = \frac{4}{4} \] This reduces to \(y^2 + \frac{z^2}{4} - x^2 = 1\). This represents a hyperboloid of one sheet.

Step 3: Sketching the Surface

To sketch \(y^2 + \frac{z^2}{4} - x^2 = 1\), start by finding traces in the coordinate planes. For the \(xy\)-plane \((z = 0)\), you have \(y^2 - x^2 = 1\), which is a hyperbola. In the \(xz\)-plane \((y = 0)\), \(-x^2 + \frac{z^2}{4} = 1\), another hyperbola. Meanwhile, for the \(yz\)-plane \((x = 0)\), \(y^2 + \frac{z^2}{4} = 1\), which is an ellipse.

Step 4: Analyzing Symmetry and Axes

The surface is symmetric about the x-axis. It opens outward in the y and z directions because of the positive coefficients of \(y^2\) and \(\frac{z^2}{4}\). Ensure your sketch shows a single 'sheet' opening along these axes resembling a double cone that was sliced horizontally.

Key Concepts

Coordinate Planes

Understanding coordinate planes is essential when sketching surfaces like an elliptic hyperboloid. Coordinate planes are imaginary flat surfaces that cut through space in three dimensions. There are three primary coordinate planes in a three-dimensional space which are:

• The xy-plane, where the z-coordinate is zero.
• The xz-plane, where the y-coordinate is zero.
• The yz-plane, where the x-coordinate is zero.

When sketching complex surfaces, examining how the surface interacts with each of these coordinate planes can provide a clearer view of the overall structure. By finding traces on these planes, as was done for the elliptic hyperboloid equation \(y^2 + \frac{z^2}{4} - x^2 = 1\), we reveal localized two-dimensional shapes such as hyperbolas and ellipses. This approach breaks down the 3D surface into simpler, more manageable parts, making it easier to understand and sketch.

Surface Sketching

Sketching a surface in 3D space might seem daunting at first, but with some practice and understanding, it becomes an illuminating exercise. For a hyperboloid, like the one described by the equation \(y^2 + \frac{z^2}{4} - x^2 = 1\), we can use the shapes identified on the coordinate planes to piece together a full picture.Steps to Surface Sketching:

• Identify the Surface Type: Recognize the structure of the equation. For instance, one negative squared term indicates a hyperboloid of one sheet.
• Determine Cross Sections: Look at each coordinate plane to find the cross-sections. The shapes (hyperbolas, ellipses) form a guideline.
• Analyze Symmetry: Note symmetrical features to replicate patterns across the figure accurately.
• Draw and Connect the Traces: Use the two-dimensional shapes obtained from the coordinate plane intersections to sketch the full surface.

This step-by-step method ensures your drawings are consistent with mathematical descriptions, easing the learning process.

Conic Sections

Conic sections arise when a plane intersects a cone in various ways, resulting in different shapes: ellipses, hyperbolas, parabolas, and circles. In the context of three-dimensional surfaces, these conic sections are crucial. They form the cross-sections of more complex surfaces, like the hyperboloid from the equation \(y^2 + \frac{z^2}{4} - x^2 = 1\).Different Conic Sections:

• Ellipse: As seen in the yz-plane trace, an ellipse is formed when the plane cuts across the cone at an angle, resulting in an oval shape.
• Hyperbola: This shape appears in both the xy-plane and the xz-plane. A hyperbola arises when the plane cuts through both halves of the cone, forming two separate curves.
• Parabola and Circle: While not applicable to this equation, these other conic sections complete the family of shapes generated by slicing cones.

Understanding these sections helps students visualize and recognize the interplay between the three-dimensional surfaces and their two-dimensional intersections. This knowledge makes the daunting task of interpreting and sketching complex geometric figures significantly more manageable.