A hollow sphere of internal and external diameters 4 cm and 8 cm respectively is melted into a cone of base diameter 8 cm. Find the height of the cone.

Sure, let's solve this step by step.

Step 1: Calculate the volume of the hollow sphere

The volume of a sphere is given by the formula (4/3)πr³. However, since this is a hollow sphere, we need to subtract the volume of the inner sphere from the volume of the outer sphere.

The radius of the outer sphere is half of the external diameter, which is 8 cm / 2 = 4 cm. The radius of the inner sphere is half of the internal diameter, which is 4 cm / 2 = 2 cm.

So, the volume of the outer sphere is (4/3)π(4 cm)³ = 268.08 cm³. And the volume of the inner sphere is (4/3)π(2 cm)³ = 33.51 cm³.

Subtracting the volume of the inner sphere from the volume of the outer sphere gives us the volume of the hollow sphere: 268.08 cm³ - 33.51 cm³ = 234.57 cm³.

Step 2: Calculate the height of the cone

The volume of a cone is given by the formula (1/3)πr²h, where r is the radius of the base and h is the height.

We know that the volume of the cone is equal to the volume of the hollow sphere, which we calculated to be 234.57 cm³. We also know that the radius of the base of the cone is half of the base diameter, which is 8 cm / 2 = 4 cm.

Substituting these values into the formula for the volume of a cone, we get:

234.57 cm³ = (1/3)π(4 cm)²h

Solving for h, we get:

h = 234.57 cm³ / [(1/3)π(4 cm)²] = 7 cm

So, the height of the cone is 7 cm.