Problem 9 - Calculus Early Transcendentals

Question

What is the name of the surface defined by the equation \(x^{2}+\frac{y^{2}}{3}+2 z^{2}=1 ?\)

Step-by-Step Solution

Answer

Answer: The name of the surface is an ellipsoid.

Step 1: Identify the coefficients and the form of the equation

Observe the given equation: \(x^{2}+\frac{y^{2}}{3}+2z^{2}=1\). The coefficients for the squares of \(x\), \(y\), and \(z\) are 1, \(\frac{1}{3}\), and 2, respectively. Since all the coefficients are positive, and there are no cross-products, we know we are dealing with an ellipsoid.

Step 2: Write the general form of an ellipsoid equation

The general equation for an ellipsoid is of the form: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\), where \(a\), \(b\), and \(c\) are the lengths of the semi-axes along the x, y, and z directions respectively.

Step 3: Determine the semi-axes lengths

Now let's determine the lengths of the semi-axes of the given equation, \(x^{2}+\frac{y^{2}}{3}+2z^{2}=1\). By comparing it with the general form of an ellipsoid equation, we can easily determine the semi-axes lengths as \(a = 1\), \(b= \sqrt{3}\), and \(c= \frac{1}{\sqrt{2}}\).

Step 4: Name the surface

Since the given equation represents an ellipsoid with semi-axes lengths \(a=1\), \(b=\sqrt{3}\), and \(c=\frac{1}{\sqrt{2}}\), the name of the surface defined by the equation \(x^{2}+\frac{y^{2}}{3}+2 z^{2}=1\) is an ellipsoid.