Final answer:
To find the height of the cone and the pyramid, use the volume and surface area formulas. The height of the pyramid is 54 cm, and the height of the cone is 18 cm.
Step-by-step explanation:
To solve this problem, we can use the formulas for the volume and surface area of a cone and a pyramid.
Let's first find the volume of the pyramid and the cone.
Given that the volume of the pyramid is three times the volume of the cone, we can set up the equation:
Volume of pyramid = 3 * Volume of cone
Let Vp be the volume of the pyramid and Vc be the volume of the cone.
Vp = 3 * Vc
Using the formula for the volume of a pyramid (Vp = (1/3) * Base Area * Height), we can substitute this into the equation:
(1/3) * Base Area * Height of pyramid = 3 * (1/3) * Base Area * Height of cone
Since the base area of the pyramid is equal to the base area of the cone, we can cancel out the terms:
Height of pyramid = 3 * Height of cone
Given that the height of the cone is 18 cm, we can substitute this into the equation:
Height of pyramid = 3 * 18 cm
Height of pyramid = 54 cm
Therefore, the height of the pyramid is 54 cm.
For the height of the cone, it is given as 18 cm.