A solid sphere is melted and recast into 18 identical cones, each having a base radius of 3 cm and height of 18 cm. Find the radius of the sphere.

The correct option is 9 cm.

To find the radius of the solid sphere, we use the principle of conservation of volume. When a solid is melted and recast into another shape, the total volume remains the same.

Therefore, the volume of the solid sphere is equal to the total volume of the 18 identical cones.

Let the radius of the sphere be $R$.

The formula for the volume of a sphere of radius $R$ is given by:

$V_{\text{sphere}}=\frac{4}{3}\pi R^3$

Let the radius of each cone be $r$ and its height be $h$. We are given:
Base radius of each cone, $r=3\text{cm}$
Height of each cone, $h=18\text{cm}$

The formula for the volume of a single cone is:

$V_{\text{cone}}=\frac{1}{3}\pi r^{2}h$

Since the sphere is recast into 18 such identical cones, we write the relation:

$V_{\text{sphere}}=18\times V_{\text{cone}}$

Substituting the volume formulas into the equation:

$\frac{4}{3}\pi R^3=18\times \left(\frac{1}{3}\pi r^{2}h\right)$

We can simplify this equation by dividing both sides by $\pi$ and multiplying both sides by 3:

$4R^3=18\times r^{2}h$

Now, substitute the given values $r=3$ and $h=18$ into the equation:

$4R^3=18\times 3^2\times 18$

$4R^3=18\times 9\times 18$

Divide both sides by 4 to solve for $R^3$:

$R^3=\frac{18\times 9\times 18}{4}$

Simplify the fraction:

$R^3=\frac{18}{2}\times 9\times \frac{18}{2}$

$R^3=9\times 9\times 9$

$R^3=9^3$

Taking the cube root on both sides:

$R=9\text{cm}$

Thus, the radius of the sphere is 9 cm.