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.

Given,

Diameter of hemisphere = 14 cm

Radius of hemisphere (r) = $\dfrac{\text{Diameter}}{2} = \dfrac{14}{2}$ = 7 cm.

From figure,

Height of cylindrical vessel (h) = 13 - 7 = 6 cm.

Radius of cylindrical vessel (R) = Radius of hemisphere (r) = 7 cm.

Inner surface area of vessel = Curved surface area of hemisphere + Curved surface area of cylinder

= 2πr² + 2πRh

= 2π(r² + Rh)

= $2 \times \dfrac{22}{7}(7^2 + 7 \times 6)$

= $\dfrac{44}{7} \times (49 + 42)$

= $\dfrac{44}{7} \times 91$

= 44 × 13

= 572 cm².

Hence, inner surface area of vessel = 572 cm².