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.) [Use\( \pi=\frac{22}{ 7}\) ]

Answer 1

Based on the provided figure:
Height of each conical section (h₁) = 2 cm
Height of the cylindrical section (h₂) = 12 − (2 × Height of conical part) = 12 − (2 × 2) = 8 cm
Radius of the cylindrical section (r) = Radius of the conical part = \(\frac{3}{2}\) cm

The total volume of air in the model is the sum of the volume of the cylinder and twice the volume of the cones.
\(Volume_{total} = Volume_{cylinder} + 2 \times Volume_{cone}\)
\(=\pi 𝑟^2ℎ_2+2×\frac{1}{ 3}\pi 𝑟^2ℎ_1\)

\(=\pi (\frac{3 }{2}) ^2 .8+2×\frac{1}{ 3}\pi (\frac{3}{ 2})^ 2 .2\)

\(=18\pi+3\pi=21\pi\)

\(=21×\frac{22}{ 7}\)
= 66 𝑐𝑚³