Directions: Solve the following problem.

A solid metallic hemisphere of radius 9 cm is melted and recast into a cone of base radius 9 cm. Find the height of the cone.

The correct option is 18 cm.

Step 1: Understand the core mathematical principle
When a solid object is melted and recast into a different shape, its volume remains constant. Therefore, the volume of the original solid metallic hemisphere must be equal to the volume of the newly formed cone.

Step 2: Identify the given dimensions
Radius of the hemisphere ($r_{\text{hemisphere}}$) = 9 cm
Base radius of the cone ($r_{\text{cone}}$) = 9 cm
Let the height of the cone be $h$.

Step 3: State the volume formulas

$\text{Volume of a hemisphere}=\frac{2}{3}\pi r^3$

$\text{Volume of a cone}=\frac{1}{3}\pi r^{2}h$

Step 4: Equate the two volumes and solve for height ($h$)

$\text{Volume of cone}=\text{Volume of hemisphere}$

$\frac{1}{3}\pi {\left(r_{\text{cone}}\right)}^{2}h=\frac{2}{3}\pi {\left(r_{\text{hemisphere}}\right)}^3$

Dividing both sides by $\frac{1}{3}\pi$:

${\left(r_{\text{cone}}\right)}^2\times h=2\times {\left(r_{\text{hemisphere}}\right)}^3$

Substitute the given radius values ($r_{\text{cone}}=9\text{cm}$ and $r_{\text{hemisphere}}=9\text{cm}$):

$9^2\times h=2\times 9^3$

Divide both sides by $9^2$:

$h=\frac{2\times 9^3}{9^2}$

$h=2\times 9$

$h=18\text{cm}$

Hence, the height of the cone is 18 cm.