A hollow sphere of external and internal diameters of 10 cm and 6 cm, respectively, is melted and made into another solid in the shape of a right circular cone of base diameter 10 cm. Find the height of the cone.

The correct answer is 15.68 cm.

When a solid is melted and recast into another shape, the volume remains conserved. So:

Volume of hollow sphere = Volume of cone

Step 1: Identify the dimensions of the hollow sphere.

External diameter = 10 cm → External radius, R = 5 cm
Internal diameter = 6 cm → Internal radius, r = 3 cm

Step 2: Calculate the volume of the hollow sphere.

The volume of a hollow sphere is:

\frac{4}{3}\pi (R^3-r^3)

Substituting R = 5 and r = 3:

=\frac{4}{3}\pi (5^3-3^3)

=\frac{4}{3}\pi (125-27)

=\frac{4}{3}\pi \times 98

=\frac{392}{3}\pi \text{cm}{}^3

Step 3: Identify the dimensions of the cone.

Base diameter of cone = 10 cm → Base radius of cone, R_c = 5 cm
Let the height of the cone = h cm

Step 4: Write the volume of the cone.

\text{Volume of cone}=\frac{1}{3}\pi R_c^{2}h

=\frac{1}{3}\pi \times 5^2\times h

=\frac{25}{3}\pi h

Step 5: Set volumes equal and solve for h.

\frac{25}{3}\pi h=\frac{392}{3}\pi

Cancelling \frac{\pi }{3} from both sides:

25h=392

h=\frac{392}{25}

h=15.68\text{cm}

∴ The height of the cone is 15.68 cm.