Question

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

Short 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-by-step solution

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.

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).

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.