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.)

Explanation

To find the volume of air contained in the model that Rachel made, we need to calculate the volume of the cylinder and the two cones attached at its ends. Here’s a step-by-step solution:

### Step 1: Identify the dimensions
- Diameter of the model = 3 cm
- Radius (r) = Diameter / 2 = 3 cm / 2 = 1.5 cm
- Total length of the model = 12 cm
- Height of each cone (h1) = 2 cm

### Step 2: Calculate the height of the cylindrical part
- The height of the cylindrical part (h2) can be calculated as:
\[ h2 = \text{Total length} - 2 \times \text{Height of each cone} = 12 \, \text{cm} - 2 \times 2 \, \text{cm} = 12 \, \text{cm} - 4 \, \text{cm} = 8 \, \text{cm} \]

### Step 3: Calculate the volume of the cylinder
- The formula for the volume of a cylinder is:
\[ V_{\text{cylinder}} = \pi r^2 h \]
- Substituting the values:
\[ V_{\text{cylinder}} = \pi (1.5 \, \text{cm})^2 (8 \, \text{cm}) = \pi (2.25 \, \text{cm}^2) (8 \, \text{cm}) = 18\pi \, \text{cm}^3 \]

### Step 4: Calculate the volume of one cone
- The formula for the volume of a cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
- Substituting the values for one cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi (1.5 \, \text{cm})^2 (2 \, \text{cm}) = \frac{1}{3} \pi (2.25 \, \text{cm}^2) (2 \, \text{cm}) = \frac{4.5}{3} \pi \, \text{cm}^3 = 1.5\pi \, \text{cm}^3 \]

### Step 5: Calculate the total volume of the two cones
- Since there are two cones:
\[ V_{\text{two cones}} = 2 \times V_{\text{cone}} = 2 \times 1.5\pi \, \text{cm}^3 = 3\pi \, \text{cm}^3 \]

### Step 6: Calculate the total volume of the model
- The total volume of the model is the sum of the volume of the cylinder and the volume of the two cones:
\[ V_{\text{total}} = V_{\text{cylinder}} + V_{\text{two cones}} = 18\pi \, \text{cm}^3 + 3\pi \, \text{cm}^3 = 21\pi \, \text{cm}^3 \]

### Step 7: Substitute the value of \(\pi\) to find the numerical value
Using \(\pi \approx \frac{22}{7}\):
\[ V_{\text{total}} = 21\pi \approx 21 \times \frac{22}{7} = \frac{462}{7} \approx 66 \, \text{cm}^3 \]

### Final Answer
The volume of air contained in the model that Rachel made is approximately **66 cm³**.

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