Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made. (Assume the outer and inner dimensions of the model to be nearly the same.)

Given,

Height of each cone (h) = 2 cm

Diameter of model (d) = 3 cm

From figure,

Radius of each cone = Radius of cylinder = r = $\dfrac{d}{2} = \dfrac{3}{2}$ = 1.5 cm

Height of cylinder (H) = 12 - 2 - 2 = 8 cm.

Volume of air inside model = Volume of cylinder + 2 × Volume of each cone

= πr²H + $2 \times \dfrac{1}{3}πr^2h$

= πr²$\Big(H + \dfrac{2}{3}h\Big)$

$= \dfrac{22}{7} \times (1.5)^2 \times \Big(8 + \dfrac{2}{3} \times 2\Big) \\[1em] = \dfrac{22}{7} \times 2.25 \times \Big(8 + \dfrac{4}{3}\Big) \\[1em] = \dfrac{22}{7} \times 2.25 \times \Big(\dfrac{24 + 4}{3}\Big) \\[1em] = \dfrac{22}{7} \times 2.25 \times \dfrac{28}{3} \\[1em] = \dfrac{1386}{21} \\[1em] = 66 \text{ cm}^3.$

Hence volume of air contained in model = 66 cm³.