Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its end by using the 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 dimension of the model to be nearly the same).

To find the volume of air contained in Rachel's model, we need to calculate the volume of the cylinder and the two cones separately, then sum them up.

Step 1: Volume of the Cylinder

The formula for the volume of a cylinder is:

V = πr²h

Where:

• r = radius of the cylinder
• h = height (or length) of the cylinder

Given the diameter is 3 cm, the radius (r) is:

r = diameter / 2 = 3 cm / 2 = 1.5 cm

The length (h) of the cylinder is 12 cm. Plugging in the values:

V_cylinder = π(1.5 cm)²(12 cm)

V_cylinder = π(2.25 cm²)(12 cm) = 27π cm³

Step 2: Volume of One Cone

The formula for the volume of a cone is:

V = (1/3)πr²h

For each cone, the radius remains the same (1.5 cm) and the height (h) is 2 cm:

V_cone = (1/3)π(1.5 cm)²(2 cm)

V_cone = (1/3)π(2.25 cm²)(2 cm) = (1/3)(4.5π cm³) = 1.5π cm³

Step 3: Total Volume of Two Cones

Since there are two cones, we multiply the volume of one cone by 2:

Total volume of cones = 2 × 1.5π cm³ = 3π cm³

Final Calculation: Total Volume of the Model

Now, we add the volume of the cylinder and the total volume of the cones:

Total Volume = V_cylinder + Total volume of cones

Total Volume = 27π cm³ + 3π cm³ = 30π cm³

Final Answer

The volume of air contained in Rachel's model is approximately:

30π cm³ or about 94.25 cm³ when using π ≈ 3.14.