A solid is in the form of a right circular cylinder with a hemispherical shape at one end and a cone at the other end. Their common diameter is 4.2 cm and the heights of the cylindrical and conical portions are 12 cm and 7 cm respectively. Find the volume of the solid.

Given,

Common diameter = 4.2 cm

Common radius (r) = $\dfrac{4.2}{2}$ = 2.1 cm.

Height of cylindrical portion (H) = 12 cm

Height of conical portion (h) = 7 cm

Volume of solid (V) = Volume of hemispherical portion + Volume of cylindrical portion + Volume of conical portion.

$V = \dfrac{2}{3}πr^3 + πr^2H + \dfrac{1}{3}πr^2h \\[1em] = \dfrac{2}{3} \times \dfrac{22}{7} \times (2.1)^3 + \dfrac{22}{7} \times (2.1)^2 \times 12 + \dfrac{1}{3} \times \dfrac{22}{7} \times (2.1)^2 \times 7 \\[1em] = 19.404 + 166.32 + 32.34 \\[1em] = 218.064 \text{ cm}^3.$

Hence, volume of solid = 218.064 cm³.