Answer
${\left( {\frac{x}{6}} \right)^2} + {\left( {\frac{y}{8}} \right)^2} = {\left( {\frac{z}{5}} \right)^2}$
Work Step by Step
When we slice the surface with $z=5$ as is shown in Figure 16(B) we get an ellipse with the semiminor axis $a=6$ located on the $x$-axis and the semimajor axis $b=8$ located on the $y$-axis. Thus, we identify that the graph of the quadric surface in Figure 16(B) is an elliptic cone as is described on page 689. It has general equation ${\left( {\frac{x}{a}} \right)^2} + {\left( {\frac{y}{b}} \right)^2} = {\left( {\frac{z}{c}} \right)^2}$.
So, from Figure 16(B) we identify that $a=6$, $b=8$, $c=5$. Thus, the equation of this elliptic cone is ${\left( {\frac{x}{6}} \right)^2} + {\left( {\frac{y}{8}} \right)^2} = {\left( {\frac{z}{5}} \right)^2}$.
