A vessel is a hollow cylinder fitted with a hemispherical bottom of the same base. The depth of the cylindrical part is $4\dfrac{2}{3}$ m and the diameter of hemisphere is 3.5 m. Calculate the capacity and the internal surface area of the vessel.

From figure,

Diameter of the base = 3.5 m

Radius of cylinder = Radius of hemispherical bottom = r = $\dfrac{3.5}{2}$ m = 1.75 m.

Height of cylindrical part (h) = $4\dfrac{2}{3} = \dfrac{14}{3}$ m.

(i) Capacity (volume) of the vessel = Volume of cylinder + Volume of hemispherical bottom

$= πr^2h + \dfrac{2}{3}πr^3 \\[1em] = πr^2\Big(h + \dfrac{2}{3}r\Big) \\[1em] = \dfrac{22}{7} \times 1.75 \times 1.75 \times \Big(\dfrac{14}{3} + \dfrac{2}{3} \times 1.75\Big) \\[1em] = 22 \times 0.25 \times 1.75 \times \Big(\dfrac{14}{3} + \dfrac{3.5}{3}\Big) \\[1em] = 9.625 \times \dfrac{17.5}{3} \\[1em] = \dfrac{168.4375}{3} \\[1em] = 56.15 \text{ m}^3.$

Internal curved surface area = Surface area of cylindrical part + Surface area of hemispherical bottom

= 2πrh + 2πr²

= 2πr(h + r)

= $2 \times \dfrac{22}{7} \times 1.75 \times \Big(\dfrac{14}{3} + 1.75\Big)$

= 2 x 22 x 0.25 x (4.67 + 1.75)

= 2 x 22 x 0.25 x 6.42

= 70.62 m².

Hence, volume of vessel = 56.15 m³ and internal curved surface area = 70.62 m².