The internal and external diameters of a hollow hemispherical vessel are 21 cm and 28 cm respectively. Find :

(i) internal curved surface area,

(ii) external curved surface area,

(iii) total surface area,

(iv) volume of material of the vessel.

(i) Given,

Internal diameter = 21 cm

Internal radius (r) = $\dfrac{21}{2}$ cm.

By formula,

Internal curved surface area = 2πr².

$= 2 \times \dfrac{22}{7} \times \dfrac{21}{2} \times \dfrac{21}{2} \\[1em] = \dfrac{19404}{28} \\[1em] = 693 \text{ cm}^2.$

Hence, internal curved surface area = 693 cm².

(ii) Given,

Internal diameter = 28 cm

Internal radius (R) = $\dfrac{28}{2}$ = 14 cm.

By formula,

External curved surface area = 2πR².

$= 2 \times \dfrac{22}{7} \times 14 \times 14 \\[1em] = 1232 \text{ cm}^2.$

Hence, external curved surface area = 1232 cm².

(iii) By formula,

Total surface area of hemisphere = 2πr² + 2πR² + π(R² - r²)

$= 693 + 1232 + \dfrac{22}{7} \times \Big[(14)^2 - \Big(\dfrac{21}{2}\Big)^2\Big] \\[1em] = 1925 + \dfrac{22}{7} \times \Big[196 - \dfrac{441}{4}\Big] \\[1em] = 1925 + \dfrac{22}{7} \times \dfrac{784 - 441}{4} \\[1em] = 1925 + \dfrac{22}{7} \times \dfrac{343}{4} \\[1em] = 1925 + \dfrac{11 \times 49}{2} \\[1em] = 1925 + \dfrac{539}{2} \\[1em] = 1925 + 269.5 \\[1em] = 2194.5 \text{ cm}^2.$

Hence, total surface area of hemisphere = 2194.5 cm².

(iv) By formula,

Volume of hemispherical vessel = $\dfrac{2}{3}π(R^3 - r^3)$

$= \dfrac{2}{3} \times \dfrac{22}{7} \times [(14)^3 - \Big(\dfrac{21}{2}\Big)^3] \\[1em] = \dfrac{2}{3} \times \dfrac{22}{7} \times [2744 - 1157.625] \\[1em] = \dfrac{2}{3} \times \dfrac{22}{7} \times 1586.375 \\[1em] = \dfrac{44}{21} \times 1586.375 \\[1em] = 3323.83 \text{ cm}^3.$

Hence, volume of vessel = 3323.83 cm³.