Answer:
The height of the cone is [tex]48\ in[/tex]
Step-by-step explanation:
step 1
Find the radius of the base of cone
we know that
The volume of the cone is equal to
[tex]V=\frac{1}{3}\pi r^{2} h[/tex]
we have
[tex]V=12,288\pi\ in^{3}[/tex]
[tex]tan(30\°)=\frac{r}{h}[/tex] ---> remember that the vertex angle of the vertical cross section is 60 degrees
so
[tex]r=(h)tan(30\°)[/tex]
[tex]r=(h)\frac{\sqrt{3}}{3}[/tex]
substitute the values and solve for h
[tex]12,288\pi=\frac{1}{3}\pi ((h)\frac{\sqrt{3}}{3})^{2} h[/tex]
[tex]36,864=\frac{h^{3}}{3}[/tex]
[tex]h^{3}=110,592[/tex]
[tex]h=48\ in[/tex]
