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A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin aluminium sheet. The diameter of the model is $3$ cm and its total length is $12$ cm. If each cone has a height of $2$ cm, find the volume of air contained in the model.
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Sol. Radius of each cone = Radius of cylinder = $\frac{3}{2}$ cm
Height of each cone 'H' = $2$ cm
Height of cylinder ‘h' = $12 – 4 = 8$ cm
Volume of air = Volume of cylinder + Volume of $2$ cones
$= \pi r^2 h + 2 \times \frac{1}{3} \pi r^2 H$
$= \pi r^2 (h + \frac{2}{3} H) = \frac{22}{7} \times (\frac{3}{2})^2 \times (8 + \frac{2}{3} \times 2)$
$= \frac{22}{7} \times \frac{9}{4} \times \frac{28}{3} = 66$ cm$^3$
Height of each cone 'H' = $2$ cm
Height of cylinder ‘h' = $12 – 4 = 8$ cm
Volume of air = Volume of cylinder + Volume of $2$ cones
$= \pi r^2 h + 2 \times \frac{1}{3} \pi r^2 H$
$= \pi r^2 (h + \frac{2}{3} H) = \frac{22}{7} \times (\frac{3}{2})^2 \times (8 + \frac{2}{3} \times 2)$
$= \frac{22}{7} \times \frac{9}{4} \times \frac{28}{3} = 66$ cm$^3$