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.)
Answers
Volume of Air Contained in Model
= Volume of Model
= Volume of Cylinder + Volume of 1st cone + Volume of 2nd Cone
= πr²h + 1/3πr²h + 1/3πr²h
= (π × r² × h) + (1/3 × π × r² × h) + (1/3 × π × r² × h)
= (22/7 × (3/2)² × h) + (1/3 × 22/7 × (3/2)² × h) + (1/3 × 22/7 × (3/2)² × h)
= (22/7 × 9/4 × h) + (1/3 × 22/7 × 9/4 × 2) + (1/3 × 22/7 × 9/4 × h)
= (198/28 × h) + (22/217 × 9/4 × h) + (22/21 × 9/4 × h)
= (198/28 × (12 - 4)) + (22/21 × 9/4 × 2) + (22/21 × 9/4 × 2)
= (198/28 × 8) + (22/21 × 9/4 × 2) + (22/21 × 9/4 × 2)
= (198/28 × 8) + (22/21 × 9/2 × 1) + (22/21 × 9/2 × 1)
= (198/28 × 8) + (22/21 × 9/2) + (22/21 × 9/2)
= (1584/28) + (198/42) + (198/42)
= (56.57) + (4.71) + (4.71)
= 65.99 cm³
≈ 66 cm³
Given,
Height of cylinder = 12–4 = 8 cm
Radius = 1.5 cm
Height of cone = 2 cm
Now, the total volume of the air contained will be = Volume of cylinder+2×(Volume of cone)
∴ Total volume = πr2h+[2×(⅓ πr2h )]
= 18 π+2(1.5 π)
= 66 cm3.