Problem 81 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The surface described by the equation $$y = 4z^2 - x^2$$ is an elliptical paraboloid.

Step 1: Rewrite the given equation

The equation for the surface is given as: $$y = 4z^2 - x^2$$

Step 2: Compare with standard equations

Compare the given equation with some well-known standard equations to identify the surface type. In this case, the equation appears to be similar to the equation of a paraboloid. To get a better understanding, we will rewrite the equation in the form of a standard paraboloid equation. The standard equation of an elliptic paraboloid is given by: $$\frac{x^2}{a^2} - \frac{y}{c} + \frac{z^2}{b^2} = 0$$

Step 3: Rewrite the given equation in standard paraboloid form

To change the given equation to the standard paraboloid form, follow these steps: 1. Move \(x^2\) and \(4z^2\) to the right side of the equation. 2. Divide and adjust the equation to match the standard form. This gives us the equation: $$y = -x^2 + 4z^2$$ $$\Rightarrow \frac{-x^2}{1} + 4\frac{z^2}{1} = -y$$ $$\Rightarrow \frac{x^2}{1} - \frac{y}{4} + \frac{z^2}{\frac{1}{4}} = 0$$ Now, comparing with the standard equation, we see that: $$a^2 = 1, b^2 = \frac{1}{4}, c = 4$$

Step 4: Describe the surface

Based on the standard paraboloid equation and the derived values for \(a, b,\) and \(c\), we can now describe the surface. We have an elliptical paraboloid centered at the origin (0, 0, 0) with its axis parallel to the y-axis, opening in the negative y-direction. The surface defined by the equation $$y = 4z^2 - x^2$$ is an elliptical paraboloid.