Final answer:
The height of the cylinder in the given solid is 2h/3, which is found by setting up an equation relating the total volume (three times the cone's volume) to the sum of the cone and cylinder volumes and solving for the cylinder's height.
Step-by-step explanation:
To solve this problem, we need to define the volume of the cone and cylinder. Given that the cone fits exactly on top of the cylinder, they share the same radius r. Since the volume of a cone is given by V(cone) = (1/3)πr²h, where h is the height of the cone, we can use this to set up an equation in terms of the cylinder's volume.
As mentioned in the question, the total volume of the solid (cylinder plus cone) is three times the volume of the cone. Mathematically, this is expressed as V(total) = 3V(cone). The volume of the solid includes the volume of the cone and the cylinder, so V(total) = V(cone) + V(cylinder).
Substituting 3V(cone) in place of V(total), we get:
3V(cone) = V(cone) + V(cylinder)
Since the volumes are equal, we can set up an equation:
2V(cone) = V(cylinder)
We know that the volume of the cylinder can be expressed as V(cylinder) = πr²H, where H is the height of the cylinder we want to find.
Let's replace V(cylinder) with its formula and V(cone) with (1/3)πr²h:
2 × (1/3)πr²h = πr²H
Cancelling out the common factors on both sides, we find:
2 × (1/3)h = H
Thus:
H = 2h/3
Therefore, the height of the cylinder is 2h/3, which corresponds to option C.