A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

Let's create a figure of the vessel according to the given description.

From the figure, it’s clear that the inner surface area of the vessel includes the curved surface area of the hemisphere and the cylinder.

The inner surface area of the vessel = CSA of the hemisphere + CSA of the cylinder

We will find the area of the vessel by using formulae;

CSA of hemisphere= 2πr², where r is the radius of the hemisphere.

CSA of the cylinder = 2πrh

where r and h are the radius and height of the cylinder respectively.

Height of the cylinder = Total height of the vessel - the height of the hemisphere.

Diameter of the hemisphere, d = 14 cm

Radius of the hemisphere, r = 14/2 cm = 7 cm

Height of the hemisphere = radius of the hemisphere, r = 7cm

Radius of the cylinder, r = 7 cm

Height of the cylinder = Total height of the vessel - height of the hemisphere

h = 13 cm - 7 cm = 6 cm

Inner surface area of the vessel = CSA of the hemisphere + CSA of the cylinder

= 2πr² + 2πrh

= 2πr (r + h)

= 2 × 22/7 × 7cm (7 cm + 6 cm)

= 2 × 22 × 13 cm²

= 572 cm²

Thus, the inner surface area of the vessel is 572 cm².