Q2:

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.)

Given:

• The model is shaped like a cylinder with two cones attached at its ends.
• Diameter of the model ($d$) = $3$ cm.
• Total length of the model ($H_{total}$) = $12$ cm.
• Height of each cone ($h_{cone}$) = $2$ cm.

To Find:

The volume of air contained in the model.

Step 1: Determine the dimensions of the cylinder and cones.

Since the diameter of the model is $3$ cm, the radius ($r$) of the cylinder and the cones is the same:

$r = \frac{d}{2} = \frac{3}{2} = 1.5$ cm.

The height of the cylinder ($h_{cyl}$) is the total length of the model minus the heights of the two cones:

$h_{cyl} = H_{total} - (2 \times h_{cone})$

$h_{cyl} = 12 - (2 \times 2) = 12 - 4 = 8$ cm.

Step 2: Formulate the total volume of the model.

The total volume of air ($V_{total}$) is the sum of the volume of the cylinder ($V_{cyl}$) and the volumes of the two cones ($2 \times V_{cone}$):

$V_{total} = V_{cyl} + 2 \times V_{cone}$

Using the formulas for volume: $V_{cyl} = \pi r^2 h_{cyl}$ and $V_{cone} = \frac{1}{3} \pi r^2 h_{cone}$

$V_{total} = \pi r^2 h_{cyl} + 2 \times \left( \frac{1}{3} \pi r^2 h_{cone} \right)$

Step 3: Calculate the volume.

Substitute the values $r = 1.5$, $h_{cyl} = 8$, and $h_{cone} = 2$:

$V_{total} = \pi (1.5)^2 (8) + \frac{2}{3} \pi (1.5)^2 (2)$

$V_{total} = \pi (2.25)(8) + \frac{2}{3} \pi (2.25)(2)$

$V_{total} = 18\pi + \frac{2}{3} \pi (4.5)$

$V_{total} = 18\pi + 2\pi (1.5)$

$V_{total} = 18\pi + 3\pi = 21\pi$

Step 4: Final numerical evaluation.

Using $\pi \approx \frac{22}{7}$:

$V_{total} = 21 \times \frac{22}{7} = 3 \times 22 = 66$ cm$^3$.

Final Answer: The volume of air contained in the model is 66 cm$^3$.