Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Identify the quadric surface. $$ z=4 x^{2}+y^{2} $$

Step-by-Step Solution

Answer

The given equation \( z=4x^2+y^2 \) represents the quadric surface of an Elliptic Paraboloid.

Step 1: Identify the standard forms of quadric surfaces

There are six standard forms of quadric surfaces: Ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets, Elliptic Cone, Elliptic Paraboloid, and Hyperbolic Paraboloid. Each type has a specific equation.

Step 2: Match the given equation to a standard form

The given equation is \( z=4x^2+y^2 \). This equation fits the form of Elliptic Paraboloid. It has the form \( z=Ax^2+By^2 \), where A and B are positive constants. In this case, A is 4 and B is 1.

Step 3: Confirmation of Elliptic Paraboloid

In an Elliptic Paraboloid, the variable z allows for an upward, basin-like shape to form. If the equation were instead \( z=-4x^2-y^2 \), this would create a downward, inverted basin shape, but because it's positive, we have an upward shape. Therefore, we can confirm that this equation models an Elliptic Paraboloid.