Final answer:
The height of the cone is 12 cm.
Step-by-step explanation:
To find the height of the cone, we need to calculate the volume of the hollow metallic sphere and then use it to find the height of the cone.
The volume of the hollow metallic sphere can be calculated using the formula V = (4/3)π(R^3 - r^3), where R is the external radius and r is the internal radius of the sphere.
Since the external diameter is 8 cm, the external radius is 4 cm. And since the internal diameter is 4 cm, the internal radius is 2 cm.
Substituting the values into the formula, we get V = (4/3)π((4^3) - (2^3)). Solving this equation gives us the volume as V = 192π cm³.
Now, let's calculate the volume of the solid cone using the formula V = (1/3)πR²h, where R is the radius of the base of the cone and h is the height of the cone.
Since the base diameter is 8 cm, the radius of the base is 4 cm. Substituting the values into the formula, we get 192π = (1/3)π(4^2)h. Solving this equation gives us the height of the cone as h = 12 cm.