Problem 17 - Essential Calculus Early Transcendentals

Question

Use traces to sketch and identify the surface. \(36 x^{2}+y^{2}+36 z^{2}=36\)

Step-by-Step Solution

Answer

The surface is an ellipsoid.

Step 1: Simplify the Equation

Start with the given equation: \( 36x^2 + y^2 + 36z^2 = 36 \). Simplify it by dividing every term by 36 to get \( x^2 + \frac{y^2}{36} + z^2 = 1 \). This equation is now in a standard form that can be analyzed.

Step 2: Recognize the Standard Form

The simplified equation \( x^2 + \frac{y^2}{36} + z^2 = 1 \) is in the form of \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \) with \( a^2 = 1 \), \( b^2 = 36 \), and \( c^2 = 1 \). This matches the form of an equation for an ellipsoid.

Step 3: Identify the Type of Surface

Since the equation matches the standard form of an ellipsoid, we can determine that the surface represented by the equation is an ellipsoid. The semi-principal axes lengths are \( a=1 \), \( b=6 \), and \( c=1 \).

Step 4: Use Traces to Sketch the Surface

To sketch, consider sections of the ellipsoid parallel to coordinate planes. When \( y = 0 \), the trace is \( x^2 + z^2 = 1 \), a circle in the \( xz \)-plane. For \( x = 0 \) and \( z = 0 \), the traces are \( \frac{y^2}{36} + z^2 = 1 \) and \( x^2 + \frac{y^2}{36} = 1 \), both ellipses. This suggests stretching on the \( y \) axis compared to the other axes.

Key Concepts

Ellipsoid

An ellipsoid is a 3D geometric surface that resembles a stretched sphere. It is defined mathematically by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). This equation describes how the shape is extended along the three axes in space. An ellipsoid has three semi-principal axes, each representing how far the surface stretches in the coordinate directions.

• The **x-axis** has a radius of \( a \).
• The **y-axis** has a radius of \( b \).
• The **z-axis** has a radius of \( c \).

In our example, the equation \( x^2 + \frac{y^2}{36} + z^2 = 1 \) represents an ellipsoid where \( a = 1 \), \( b = 6 \), and \( c = 1 \). Here, the elongated part is along the y-axis, making it look more oval in that direction while being spherical along the x and z axes.

Traces in 3D

Traces are cross-sections of a 3D surface that help illustrate its shape. They are like the shadows a 3D object would cast in each coordinate plane. To find traces, set one variable to a constant value and solve for the others. This gives us a 2D figure representing the intersection of the ellipsoid with a plane parallel to one of the axes.

• **Trace in the xz-plane**: Setting \( y = 0 \), the equation becomes \( x^2 + z^2 = 1 \). This is a circle, showing that in this plane, the ellipsoid looks circular.
• **Trace in the yz-plane**: Setting \( x = 0 \), we have \( \frac{y^2}{36} + z^2 = 1 \). This is an ellipse with the main axis along the y-axis.
• **Trace in the xy-plane**: Setting \( z = 0 \), the equation \( x^2 + \frac{y^2}{36} = 1 \) also forms an ellipse, underscoring the elliptical nature along the y-axis.

These traces help visualize the 3D shape by revealing its 2D appearances across different slices.

Coordinate Planes

Coordinate planes are planes that divide a 3D space into different sections. They are critical in helping us understand and visualize complex surfaces like ellipsoids. The three main coordinate planes in the Cartesian system are:

• **xz-plane**: Formed by keeping y constant (usually set to zero). This plane is vertical and shows us how objects appear when only considering the x and z directions.
• **yz-plane**: Formed by keeping x constant. This is another vertical plane providing a view of the y and z directions.
• **xy-plane**: Created by setting z constant, offering a horizontal look at the x and y axes.

Each coordinate plane helps provide a simpler understanding of a complex shape by allowing us to consider one dimension less at a time. In the context of our ellipsoid, analyzing traces in each of these coordinate planes offers a clearer composite picture of the ellipsoid's shape and orientation.