Q) A student was asked to make a model 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.


A student was asked to CBSE Class 10th Board Important Questions NCERT Exemplar IGCSE ICSE IB CISCE NIOS

We know that, volume of the cone = \frac{1}{3}\pi r2 h1. Here, r is the radius of cylinder and h1 is the height of cylinder.

and the volume of the cylinder = \pi r2 h2. Here, r is the radius of cone and h2 is the height of cone.

[Note: Since radius is same for given cylinder and both cones, we have taken same variable of r, but since heights are different for both shapes, we have taken different variables.]

The volume of the model = volume of left cone + volume of the cylinder + volume of right cone

= \frac{1}{3}\pi r2 h1 + \pi r2 h2 + \frac{1}{3}\pi r2h1

= \pi r2 ( \frac{1}{3} h1 + h2 + \frac{1}{3} h1)

= \pi r2 ( \frac{2}{3} h1 + h2)

It is given that the diameter of the model is 3 cm

\therefore the radius r = \frac{3}{2} cm;

the height of cone h1 = 2 cm;

the height of the cylinder h2 =  12 – 2 – 2 = 8 cm

Hence, the volume of the model:

= (\frac{22}{7}) (\frac{3}{2})^2 [(\frac{2}{3})(2) + 8]

= (\frac{22}{7}) (\frac{9}{4}) (\frac{28}{3})

= 22 x 3 = 66 cm3

Therefore, the volume of the model is 66 cm3

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