Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that the projection in the \(x z\)-plane of the curve that is the intersection of the surfaces \(y=4-x^{2}\) and \(y=x^{2}+z^{2}\) is an ellipse, and find its major and minor diameters.

Step-by-Step Solution

Answer

The projection is an ellipse with major diameter 4 and minor diameter approximately 2.83.

Step 1: Identify the Given Equations

We are given two surface equations: 1. The first surface is described by the equation \( y = 4 - x^2 \).2. The second surface is described by the equation \( y = x^2 + z^2 \). These surfaces intersect along a curve.

Step 2: Find the Curve of Intersection

To find the intersection of the two surfaces, we set their equations equal to each other: \[ 4 - x^2 = x^2 + z^2. \] Simplifying this, we have:\[ 4 = 2x^2 + z^2. \] Rearranging gives the equation of the curve in terms of \( x \) and \( z \):\[ z^2 = 4 - 2x^2. \]

Step 3: Rewrite Equation in Standard Form

We want to identify the type of conic section represented by the equation \( z^2 = 4 - 2x^2 \). Rearrange the terms:\[ \frac{x^2}{2} + \frac{z^2}{4} = 1. \] This is the standard form of an ellipse.

Step 4: Identify Major and Minor Axes

The equation \( \frac{x^2}{2} + \frac{z^2}{4} = 1 \) is in the form of an ellipse equation \( \frac{x^2}{a^2} + \frac{z^2}{b^2} = 1 \), where \( a^2 = 2 \) and \( b^2 = 4 \). Here, \( b > a \), indicating \( z \)-axis is the major axis.- Major diameter is \( 2b = 2\sqrt{4} = 4 \).- Minor diameter is \( 2a = 2\sqrt{2} \approx 2.83 \).

Key Concepts

Intersection of Surfaces

When two three-dimensional surfaces, like those in our problem, meet, they form a curve at their intersection.
The exercise provides two different surfaces described by the equations:

• The first surface: \( y = 4 - x^2 \)
• The second surface: \( y = x^2 + z^2 \)

To figure out where these surfaces intersect, we set the equations equal to each other:

• \(4 - x^2 = x^2 + z^2\)

This step is crucial because it simplifies to \(4 = 2x^2 + z^2\).

Understanding this means recognizing how shapes meet in space, revealing a two-dimensional curve of intersection that contains points from both surfaces.

Conic Sections

Conic sections are shapes created by slicing a cone with a plane.
These sections include circles, ellipses, parabolas, and hyperbolas. In this exercise, the situation is about identifying the shape of the curve within the \(x z\)-plane, derived from the intersection equation \(z^2 = 4 - 2x^2\).
This equation is key to determining the type of conic section we have.

• In simpler terms, the way the plane intersects the double-napped cone generates different curves.
• An ellipse is formed when the intersection angle results in a closed, bounded shape.

Recognizing the equation's form as \(\frac{x^2}{2} + \frac{z^2}{4} = 1\) shows that the curve is an ellipse.

Major and Minor Axes

Every ellipse includes two essential features: the major and minor axes.

• The major axis is the longest part of the ellipse; think of it as the ellipse's long stretch.
• The minor axis, shorter, spans the widest part perpendicular to the major axis.

In our problem, the standard ellipse equation \(\frac{x^2}{2} + \frac{z^2}{4} = 1\) helps us locate these axes by comparing values:

• Since \(b^2 = 4\) and \(a^2 = 2\), we note that the \(z\)-axis is major, having a length of 4.
• The \(x\)-axis is minor, with a length approximately 2.83.

These axes help define the ellipse's size and orientation within the \(x z\)-plane.

Ellipse Equation

Understanding the ellipse equation is fundamental to describe an ellipse precisely.
It usually takes the form: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where:

• \(a\) is the semi-major axis, indicating the half-length along the longest diameter.
• \(b\) is the semi-minor axis, indicating the half-length along the shortest diameter.

For our ellipse \(\frac{x^2}{2} + \frac{z^2}{4} = 1\), \(a\) equals \(\sqrt{2}\) and \(b\) equals \(2\).
These values represent the radii in the \(x\) and \(z\)-directions.
This equation gives precise measurements, helping interpret the ellipse's shape and position in the plane.