A cone has a volume of 108 pi and a base diameter of 12. What is the height of the cone?

h = 9 units of length

Step-by-step explanation:

Volume of a circular right cone is:

V (c) = (1/3) * Area of the circular base * h

Where h is the height of the cone, then

V (c) = (1/3) * π * r² * h

And we have that V (c) = 108*π

Then:

108 * π = (1/3) * π * r² * h (1)

We know diameter of the circular base is 12 units, according to that

r = d/2 ⇒ r = 12/2 r = 6 units

Then:

108 * π [ cubic units ] = (1/3) * π * (6) ² * h [square units] [ units of length]

When simplifying we have to take into account that in order to keep last equation h needs to have units of length

Therefore:

108 = (1/3) * 36 * h

h = (108 * 3) / 36

h = 9 units of length