Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+36 y^{2}=144 $$

Step-by-Step Solution

Answer

The graph is a cylinder in three-space with an elliptical cross-section in the xy-plane.

Step 1: Standard Form Identification

The given equation is \(4x^2 + 36y^2 = 144\). Compare it with the general form of an ellipse in the \(xy\)-plane: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Our equation needs to be rewritten in this format to easily identify it as the graph of an ellipse.

Step 2: Rewrite in Standard Form

First, divide the entire equation by 144 to obtain \(\frac{4x^2}{144} + \frac{36y^2}{144} = 1\). Simplify this to \(\frac{x^2}{36} + \frac{y^2}{4} = 1\). Now the equation is in the standard form of an ellipse, where \(a^2 = 36\) and \(b^2 = 4\).

Step 3: Determine the Axes Lengths

From \(\frac{x^2}{36} + \frac{y^2}{4} = 1\), calculate the axes lengths: \(a = \sqrt{36} = 6\) and \(b = \sqrt{4} = 2\). This means the ellipse has a major axis of length 12 along the x-axis (since \(a > b\)) and a minor axis of length 4 along the y-axis.

Step 4: Sketch the Graph

Draw a horizontal x-axis and a vertical y-axis. Mark the distances from the origin along the x-axis as \(\pm 6\) and along the y-axis as \(\pm 2\). Connect these points in an elliptical shape centered at the origin. This represents the ellipse in the \(xy\)-plane.

Step 5: Extend to Three-Space

Since the equation involves only \(x\) and \(y\), it represents a cylinder parallel to the z-axis (extending infinitely along the z direction). The ellipse described is the cross-section of this cylinder in the \(xy\)-plane.

Key Concepts

Ellipse

An ellipse is a geometric shape that looks like a flattened circle. It is defined by its standard form equation in the two-dimensional plane:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]In this equation:

• \(a\) represents the semi-major axis, which is half the longest diameter of the ellipse.
• \(b\) is the semi-minor axis, which is half the shortest diameter.
• The ellipse is centered at the origin if no shifts are present in the equation.

To find the axes' lengths, take the square root of the denominators: \(a = \sqrt{a^2}\) and \(b = \sqrt{b^2}\). If \(a > b\), the major axis lies along the \(x\)-axis; if \(b > a\), it lies along the \(y\)-axis.
Ellipses are commonly found in astronomy, representing orbits, and in various engineering applications.

Standard Form

The standard form of an ellipse's equation is crucial for easily identifying and sketching the ellipse. It helps determine its key features, such as the lengths of the axes and center location.To write an ellipse equation in standard form:

• Start by making sure the equation is set to 1 on one side.
• Divide the entire equation by the number on the right side of the equation.
• Simplify the fractions to match the standard form:

For example, consider the equation \(4x^2 + 36y^2 = 144\). By dividing each side by 144:\[ \frac{4x^2}{144} + \frac{36y^2}{144} = 1 \]This simplifies to:\[ \frac{x^2}{36} + \frac{y^2}{4} = 1 \]This manipulation allows us to identify \(a^2 = 36\) and \(b^2 = 4\), making it straightforward to find the semi-axes lengths and understand the ellipse's orientation.

Cylinder in Three-Space

A cylinder in three-dimensional space, often referred to as three-space, can take various forms. In this context, it refers to extending a known shape, like an ellipse, along another dimension without altering the cross-section.Here's how it works for an ellipse in the \(xy\)-plane:

• The equation \(\frac{x^2}{36} + \frac{y^2}{4} = 1\) describes an ellipse in two dimensions.
• Extending this along the \(z\)-axis (with no \(z\) term in the equation) creates a cylinder.
• This cylinder is consistent in its elliptical cross-section along the entire \(z\)-axis.

Think of it as stacking many ellipses from the \(xy\)-plane along the \(z\)-axis. The cylinder extends infinitely unless bounded by additional surfaces or constraints.
This concept is vital in fields like architecture and design, where structures extend shapes into three dimensions for practical use.