Question

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.)

• A. 50 $c{m}^3$
• B. 66 $c{m}^3$
• C. 62 $c{m}^3$
• D. 75 $c{m}^3$

Answer 1

The correct option is B

66 $c{m}^3$

Diameter of Cylinder = Diameter of Cones = 3 cm

$\Rightarrow$ Radius of Cylinder = Radius of Cones $=r=\frac{3}{2}=1.5cm$

Height of each cone $=h_1=2cm$

Length of model = 12 cm

Height of Cylinder, h
= Total Height of Model - height of 1st Cone - height of 2nd Cone
$=12-2-2=8cm$

Volume of air in the model
= Volume of Solid
= Volume of Cylinder + Volume of 1st Cone + Volume of 2nd Cone

We know that, volume of a cylinder of radius r and height h is $\pi r^{2}h$ and volume of a cone of radius R and height H is given by $\frac{1}{3}\pi R^{2}H.$

Therefore, volume of air in the model
$=\pi r^{2}h+\frac{1}{3}\pi r^2{h}_1+\frac{1}{3}\pi r^2{h}_1$

$=\pi r^2(h+\frac{h_1}{3}+\frac{h_1}{3})$

$=\frac{22}{7}\times 1.5\times 1.5(8+\frac{4}{3})$

$=\frac{22}{7}\times 1.5\times 1.5\left(\frac{28}{3}\right)=66c{m}^3$