A solid lies between two planes perpendicular to the $x-$axis at $x=0$ and $x=24.$ The cross$-$sections by planes perpendicular to the $x-$axis are circular disks whose diameters run from the line $y=\frac{x}{2}$ to the line $y=x$ as shown in the accompanying figure. Explain why the solid has the same volume as a right circular cone with base radius $6$ and height $24.$

The radius of a circular cross section of the solid at any value of $x$ is $\frac{x}{4}.$ The height of the solid is $24.$

Locate the right circular cone with base radius $6$ and height $24$ so that its vertex is at the origin and its height is along the $x-$axis. This cone is the surface of revolution of the function $$ about the $x-$axis.

$($Type an equation.$)$