Problem 83 - Calculus Early Transcendentals

Question

Identify and briefly describe the surfaces defined by the following equations. $$y=x^{2} / 6+z^{2} / 16$$

Step-by-Step Solution

Answer

Question: Describe the surface represented by the equation $$y=\frac{x^2}{6}+\frac{z^2}{16}$$, and state its main features. Answer: The surface is an elliptic paraboloid with its vertex at the origin (0, 0, 0). It is symmetric around the y-axis and opens upward along the positive y-axis.

Step 1: Identify the type of surface

We can rewrite the given equation as: $$\frac{x^2}{6} + \frac{z^2}{16} = y$$ This equation is similar to the general equation of an elliptic paraboloid: $$\frac{x^2}{a^2} + \frac{z^2}{c^2} = y$$ So, the surface defined by the given equation is an elliptic paraboloid.

Step 2: Determine the main features of the elliptic paraboloid

The main features of an elliptic paraboloid include the vertex, the axis of symmetry, and the direction of the opening. 1. Vertex: The vertex of the elliptic paraboloid is the point where the surface touches or crosses the plane y = 0. In our case, this occurs when x = 0 and z = 0. Therefore, the vertex is at the origin, V(0, 0, 0). 2. Axis of symmetry: The axis of symmetry is the line passing through the vertex and perpendicular to the plane of the paraboloid's base ellipse. In this case, the axis is the y-axis, along which it exhibits rotational symmetry. 3. Direction of the opening: Since the constants a^2 and c^2 in the equation are both positive and y is on one side of the equation, the paraboloid opens upward (along the positive y-axis).

Step 3: Briefly describe the surface

The surface defined by the equation $$y=\frac{x^2}{6}+\frac{z^2}{16}$$ is an elliptic paraboloid. Its vertex is at the origin (0, 0, 0), it is symmetric around the y-axis, and it opens upward along the positive y-axis.