A hemispherical storage vessel having a total surface area of $1323\pi$ sq cm is filled to capacity with cooking oil. The entire quantity of oil is transferred into a cylindrical drum of the same radius, filling it completely. What is the height of the cylindrical drum?

The correct option is 14 cm.

Step 1: Find the radius of the hemispherical storage vessel.

The total surface area of a hemisphere of radius $r$ is given by the formula:

Total Surface Area = $3\pi r^2$

Given that the total surface area is $1323\pi {\text{cm}}^2$:

$3\pi r^2=1323\pi$

Dividing both sides by $3\pi$:

$r^2=441$

Taking the square root of both sides:

$r=\sqrt{441}=21\text{cm}$

Step 2: Relate the volume of the hemispherical vessel to the cylindrical drum.

The volume of cooking oil in the hemispherical vessel filled to capacity is:

Volume of Hemisphere = $\frac{2}{3}\pi r^3$

Since the entire quantity of oil is transferred into a cylindrical drum of the same radius $r$ and height $h$, filling it completely, the volume of the cylindrical drum equals the volume of the hemisphere:

Volume of Cylinder = $\pi r^{2}h$

Equating the two volumes:

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

Dividing both sides by $\pi r^2$ simplifies to:

$h=\frac{2}{3}r$

Step 3: Calculate the height of the cylindrical drum.

Substitute $r=21\text{cm}$ into the height expression:

$h=\frac{2}{3}\times 21=14\text{cm}$

Thus, the height of the cylindrical drum is 14 cm.