Question

A cone has a volume of $12288\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. what is the height of the cone? express your answer as a decimal to the nearest tenth.

Answer 1

The height of the cone is
`48\ in`

Explanation:

step 1

Find the radius of the base of cone

we know that

The volume of the cone is equal to

`V=(1)/(3)\pi r^(2) h`

we have

`V=12,288\pi\ in^(3)`

`tan(30\°)=(r)/(h)` ---> remember that the vertex angle of the vertical cross section is 60 degrees

so

`r=(h)tan(30\°)`

`r=(h)(√(3))/(3)`

substitute the values and solve for h

`12,288\pi=(1)/(3)\pi ((h)(√(3))/(3))^(2) h`

`36,864=(h^(3))/(3)`

`h^(3)=110,592`

`h=48\ in`