Question

In Exercises \(11-18,\) identify and sketch the quadric surface. Use a computer algebra system to confirm your sketch. $$ 16 x^{2}+9 y^{2}+16 z^{2}-32 x-36 y+36=0 $$

Short answer

The given equation represents an ellipsoid with semi-principal axes along the x, y and z axis with lengths 2, 3 and 2, respectively.

Step-by-step solution

Rearrange the equations into Standard Form

Rewrite the given equation to standard form by completing the square and consolidating like terms. This can be done as follows: \[16 (x^{2}-2x+1)+9 (y^{2}-4y+4)+16z^{2}=1+4+36 (remove the constants on the right side, completing the square)\]therefore, \[\left(\frac{x-1}{2}\right)^{2}+\left(\frac{y-2}{3}\right)^{2}+\left(\frac{z}{2}\right)^{2}=1. This is an equation of an ellipsoid.

Sketch the surface

This takes the form of (x/a)^2 + (y/b)^2 + (z/c)^2 = 1, which is a form of an ellipsoid where a, b and c are the lengths of its semi-principal axes. Sketching this involves understanding that the semi-principal axes are along the x, y and z axis with lengths 2, 3 and 2, respectively. Hence draw an 3D oval shape with the specified proportions.

Use a computer algebra system to confirm the sketch

Confirm this sketch by inserting the equation of the surface into a computer algebra software. The software should display a 3D graph that can be rotated and viewed from multiple angles. It should match the hand-drawn sketch.