A cylinder with a diameter of 24 inches and a height of 8 inches has the same volume as a cone with a diameter of 12 inches. What is the height of the cone?

To find the height of the cone, we first calculate the volume of the cylinder and then set it equal to the volume of the cone.

1. Volume of the Cylinder (V_cylinder):

The formula for the volume of a cylinder is:

[

V = pi r^2 h

]

For the cylinder:

– Diameter = 24 inches, so radius ( r = 12 ) inches.

– Height ( h = 8 ) inches.

[

V_{cylinder} = pi (12^2)(8) = pi (144)(8) = 1152pi text{ cubic inches}

]

2. Volume of the Cone (V_cone):

The formula for the volume of a cone is:

[

V = frac{1}{3} pi r^2 h

]

For the cone:

– Diameter = 12 inches, so radius ( r = 6 ) inches.

– We don’t know the height ( h ) yet.

[

V_{cone} = frac{1}{3} pi (6^2)h = frac{1}{3} pi (36)h = 12pi h

]

3. Setting the volumes equal:

Since the volumes