A solid metal sphere of radius 6 cm is completely immersed in a vertical cylindrical vessel containing water , causing the water level to rise by 4 cm. What is the radius of the cylinder?

The correct option is $6\sqrt{2}$ cm.

Step 1: Understand the Principle
When a solid sphere is completely immersed in water inside a cylindrical vessel, the volume of water displaced is equal to the volume of the sphere. This displaced water causes the water level in the cylindrical vessel to rise.

Step 2: Formulas for Volume
1. The volume of a sphere of radius _rs is given by:

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

2. The volume of the displaced water in the cylinder of radius _R and height of water rise _h is given by:

$V_{\text{cylinder}}=\pi R^{2}h$

Step 3: Substitute the Given Values
Radius of the sphere, _{rs = 6 cm}
Rise in water level in the cylinder, _{h = 4 cm}

Equating the volume of the sphere to the volume of the displaced water in the cylinder:

$\pi R^{2}h=\frac{4}{3}\pi r_{\text{s}}^3$

Cancel _π from both sides:

$R^2\times 4=\frac{4}{3}\times 6^3$

Divide both sides by 4:

$R^2=\frac{1}{3}\times 216$

$R^2=72$

Step 4: Solve for the Radius R

$R=\sqrt{72}$

Simplify _$\sqrt{72}$:

$R=\sqrt{36\times 2}=6\sqrt{2}\text{ cm}$

Thus, the radius of the cylinder is $6\sqrt{2}$ cm.