Problem 11 - Thomas Calculus

Question

In Exercises \(1-12,\) match the equation with the surface it defines. Also, identify each surface by type (paraboloid, ellipsoid, etc.) The surfaces are labeled (a ) \(-(1)\) . $$ x^{2}+4 z^{2}=y^{2} $$

Step-by-Step Solution

Answer

The equation describes a double cone (cone surface type).

Step 1: Identify the General Form

Look at the given equation: \(x^2 + 4z^2 = y^2\). Notice that the equation involves squares of terms. This indicates that it may be related to a quadratic form, potentially describing something like an ellipsoid, hyperboloid, or paraboloid.

Step 2: Rearrange the Equation

Rearrange the equation so that it resembles a known form. The current equation is \(x^2 + 4z^2 = y^2\), which can be rewritten as \(x^2 - y^2 + 4z^2 = 0\).

Step 3: Analyze the Characteristics

The rearranged equation \(x^2 - y^2 + 4z^2 = 0\) is a form of the general quadratic expression used for hyperbolic surfaces. Specifically, it has one positive and one negative square term involving \(x\) and \(y\).

Step 4: Identify the Surface Type

Based on the equation \(x^2 - y^2 + 4z^2 = 0\), this is recognized as a type of cone surface, often referred to as a double cone. This type of equation is typically identified with a hyperboloid configuration when set to zero.

Key Concepts

Paraboloid

A paraboloid is a surface that can be described by a quadratic equation, which usually involves one squared term and two linear ones. This makes them different from more familiar shapes that only involve squared terms. Paraboloids can be classified into two types:

• Elliptic Paraboloid: Resembles a bowl and is defined by an equation like \( z = x^2 + y^2 \). The cross-sections parallel to the x-y plane are ellipses, while those parallel to one axis are parabolas.
• Hyperbolic Paraboloid: Known for its saddle shape, the equation is similar to \( z = x^2 - y^2 \). Here, cross-sections show parabolic and hyperbolic curves.

Paraboloids are prevalent in fields like architecture and physics, particularly in satellite dishes and reflector telescopes. They naturally focus parallel beams of light or energy to a single point.

Ellipsoid

Ellipsoids are symmetrical shapes that revolve around one or more of their axes, resembling elongated spheres. They can be vertically stretched or flattened, depending on their defining equation, which usually looks like:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]Ellipsoids can often be found in nature and art:

• Prolate Ellipsoid: Longer along one axis, similar to a stretched sphere or rugby ball.
• Oblate Ellipsoid: Flatter at the poles, like the shape of the Earth or a lentil.

Ellipsoids are widely helpful, from designing lenses and gears to understanding atomic structures and gravitational effects.

Hyperboloid

Hyperboloids are fascinating quadric surfaces, characterized usually by their saddle-like appearance. Their equations often contain terms like \( x^2, y^2, \) and \( z^2 \), and they take forms like:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\]There are two main types:

• One-Sheeted Hyperboloid: Looks similar to a cooling tower or an hourglass, and is formed when two of its quadratic terms are positive and one is negative.
• Two-Sheeted Hyperboloid: Seen as two separate surfaces and occurs when one term is positive and the others are negative.

These forms constantly appear in engineering and architecture because of their strength and aesthetic appeal, such as structures like observatory domes and hyperbolic cooling towers.

Cone

Cones are another common quadric surface with a distinct point called the apex and a circular base. A simple, familiar cone is defined mathematically by equations like \( x^2 + y^2 = z^2 \).Interestingly, the equation \( x^2 - y^2 + 4z^2 = 0 \) given in the original problem actually forms a cone, often termed as a double cone, due to the squared negative term involvement.The cone's analytical form can sometimes be tricky to visualize, but they can be broken down into:

• Right Circular Cone: A cone with a circular base and the apex directly above the center of the base.
• Oblique Cone: When the apex is not aligned with the center of the base, creating a slanted angle.

Recognizing these surfaces helps in understanding natural phenomena and man-made structures like water towers or megaphone designs.