Problem 85 - Calculus Early Transcendentals

Question

Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$

Step-by-Step Solution

Answer

Identify the type of surface represented by the equation 9x^2 + y^2 - 4z^2 + 2y = 0 and provide a brief description of it. The given equation represents a hyperboloid of two sheets. The surface consists of two separate, sheet-like structures, oriented along the y-axis, with principal axes along the x-, y-, and z-axes.

Step 1: Identify and group similar terms

We are given: $$9x^2 + y^2 - 4z^2 + 2y = 0$$ Rearrange the terms, grouping similar terms together: $$9x^2 + (y^2 + 2y) - 4z^2 = 0$$ ##Step 2: Complete the square##

Step 2: Complete the square for the y terms

We can complete the square for the y terms by adding and subtracting \(\frac{1}{2^2}\) inside the parentheses: $$(y^2 + 2y + 1) - 1 = (y + 1)^2 - 1$$ Now, rewrite the equation using the completed square for the y terms: $$9x^2 + ((y + 1)^2 - 1) - 4z^2 = 0$$ ##Step 3: Identify the type of surface##

Step 3: Compare the equation to the standard forms of quadratic surfaces

We have now rewritten the equation in a more recognizable form: $$9x^2 + (y + 1)^2 - 1 - 4z^2 = 0$$ Compare this equation to the standard forms of quadratic surfaces (ellipsoid, hyperboloid of one/two sheet(s), elliptic cone/paraboloid, and hyperbolic paraboloid). Notice that this equation has a positive x term, a positive y term, and a negative z term. This matches the standard form of a hyperboloid of two sheets: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$$ In our case, \(a^2 = \frac{1}{9}\), \(b^2 = 1\), and \(c^2 = 4\). ##Step 4: Describe the surface##

Step 4: Briefly describe the hyperboloid

The equation represents a hyperboloid of two sheets with principal axes along the x-, y-, and z-axes. The surface will consist of two separate, sheet-like structures, oriented along the y-axis (since the y term has a positive coefficient).