Let's denote the radius of both the cylinder and the cone as "r" and the height of the cone as "h". We know that the volume of a cylinder is given by:
V_cylinder = πr^2h
We also know that the volume of a cone is given by:
V_cone = (1/3)πr^2h
Since the cylinder and the cone have the same volume, we can set these two equations equal to each other and solve for "h":
πr^2h = (1/3)πr^2h
Simplifying the equation by dividing both sides by πr^2, we get:
h = (1/3)h
Multiplying both sides by 3, we get:
3h = h
Subtracting "h" from both sides, we get:
2h = 0
Dividing both sides by 2, we get:
h = 0
This is a non-sensical result, so we must have made an error in our calculations. The mistake occurred when we divided both sides of the equation by πr^2. Since the radius is the same for both the cylinder and the cone, it cancels out of the equation, leaving us with:
h_cylinder = h_cone
In other words, the height of the cone must be the same as the height of the cylinder, which is given as six feet. Therefore, the height of the cone is also six feet.