Problem 82 - Calculus Early Transcendentals

Question

Identify and briefly describe the surfaces defined by the following equations. $$-y^{2}-9 z^{2}+x^{2} / 4=1$$

Step-by-Step Solution

Answer

Question: Identify and describe the surface represented by the equation: $$-y^2 -9z^2 + \frac{x^2}{4}=1$$ Answer: The given equation represents a hyperboloid of one sheet, with a saddle-like shape opening in both the x and y directions, and it is symmetric about all three axes.

Step 1: Analyze the given equation

The equation given is: $$-y^{2}-9 z^{2}+x^{2} / 4=1$$ We will analyze each term and the signs of the coefficients to help classify the surface.

Step 2: Classify the surface

Looking at each term, we have \(x^2\), \(y^2\), and \(z^2\). All three variables have squared terms, and the coefficients are different. Also, note that the sign of the coefficient for \(x^2\) is positive, while the signs of the coefficients for \(y^2\) and \(z^2\) are negative. This combination of features indicates that the given equation represents a hyperboloid of one sheet.

Step 3: Describe the surface

The surface represented by the given equation, $$-y^2 -9z^2 + \frac{x^2}{4}=1$$, is a hyperboloid of one sheet. Its main features are: - It has a saddle-like shape that opens in two directions (along the x-axis and the y-axis). - Since the coefficient of \(z^2\) is much larger than the coefficient of \(y^2\), the surface will open wider in the z-direction. - The surface is symmetric about the x-axis, y-axis, and z-axis. In conclusion, the given equation represents a hyperboloid of one sheet, with a saddle-like shape opening in both the x and y directions, and it is symmetric about all three axes.