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 aluminum sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, then what will be 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.)?

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cm 3

cm 3

cm 3

cm 3

answer is B.

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Detailed Solution

It is given Diameter of the model is 3cm and Length of the model is 12cm and height of the cone is 2 cm.

Question Image

Now, from the figure;
Length of the model

= H e i g h t o f t h e c y l i n d r i c a l p a r t + 2 × H e i g h t o f t h e c o n i c a l p a r t

Height of the cylindrical part

= L e n g t h o f t h e m o d e l − 2 × H e i g h t o f t h e c o n i c a l p a r t

Consider h1 be the height of the cylindrical part.

Now, Diameter of the model, d = 3cm
Radius of cylindrical part = radius of conical part

⇒ r = 32 ⇒ r = 1.5 cm

Volume of the cone

= 13 π r 2 h 2

Volume of the cylinder

= π r 2 h 1

Now, Volume of air in the model is;

⇒ 2 × V o l u m e o f t h e c o n i c a l p a r t s + V o l u m e o f t h e c y l i n d r i c a l p a r t

⇒ 2 × 13 π r 2 h 2 + π r 2 h 1

Substituting the values of r, h1 and h2, we get

⇒ π r 2 23 h 2 + h 1 ⇒ 227 × 1.5 × 1.5 × 23 × 2 + 8 cm 3 ⇒ 227 × 1.5 × 1.5 × 283 cm 3 ⇒ 66 cm 3

Therefore, the volume of air in the model is

66 cm 3

.
Hence option (2) is correct.

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