Calculate the radius of the metal ball

Given the diameter of the solid metal ball is $6\text{ cm}$, the radius ($r$) of the ball is half of the diameter.

$r = \frac{6}{2} = 3\text{ cm}$

Equate the volume of the sphere and cylinder

When a solid metal ball is melted and recast into a solid cylinder, its volume remains conserved. Therefore, Volume of Sphere = Volume of Cylinder.

$\frac{4}{3}\pi r^3 = \pi R^2 h$

Substitute the given radii

Given that the diameter (and hence the radius) of the base of the cylinder is the same as that of the ball, we have $R = r = 3\text{ cm}$. Substituting $r = 3$ and $R = 3$ into the volume equation:

$\frac{4}{3}\pi (3)^3 = \pi (3)^2 h$

Solve for height ($h$)

Simplifying the equation: $\frac{4}{3} \times 27 = 9 \times h \implies 36 = 9h \implies h = \frac{36}{9} = 4\text{ cm}$.

$h = 4\text{ cm}$