Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Identify the quadric surface. $$ \frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1 $$

Step-by-Step Solution

Answer

The given quadric surface equation represents an ellipsoid.

Step 1: Identify the form of the equation

Identify which type of standard quadric surface equation the given equation matches. Most quadric surfaces can be expressed in a standard form, where the squared variables are all added or subtracted and each divided by a constant squared. So, looking at the given equation \(\frac{x^{2}}{9}+\frac{y^{2}}{16}+\frac{z^{2}}{16}=1\) , all variables have degree 2, all are added together and are each divided by a squared constant, which is the standard form for an ellipsoid.

Step 2: Confirming the identification of the quadric surface

Now, confirm whether our identification matches the definition of an ellipsoid. By definition, an ellipsoid is a quadric surface for which all squared terms are added together and their coefficients are all positive. This matches exactly the provided equation.