A cylindrical tube, open at both ends, is made of metal. The internal diameter of the tube is 10.4 cm and its length is 25 cm. The thickness of the metal is 8 mm everywhere. Calculate the volume of the metal.

Explanation

To calculate the volume of the metal in the cylindrical tube, we need to follow these steps:

### Step 1: Convert all measurements to the same unit
The internal diameter of the tube is given as 10.4 cm, and the thickness of the metal is given as 8 mm. We need to convert the thickness from millimeters to centimeters for consistency.

1 cm = 10 mm
So, 8 mm = 8/10 cm = 0.8 cm

### Step 2: Calculate the internal radius
The internal diameter is 10.4 cm, thus the internal radius (r) can be calculated as:
\[ r = \frac{\text{internal diameter}}{2} = \frac{10.4 \text{ cm}}{2} = 5.2 \text{ cm} \]

### Step 3: Calculate the external radius
The thickness of the metal is 0.8 cm, so the external radius (R) can be calculated as:
\[ R = r + \text{thickness} = 5.2 \text{ cm} + 0.8 \text{ cm} = 6.0 \text{ cm} \]

### Step 4: Calculate the volume of the outer cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi R^2 h \]
Where:
- \( R \) is the external radius
- \( h \) is the height (length) of the cylinder

Substituting the values:
\[ V_{\text{outer}} = \pi (6.0 \text{ cm})^2 (25 \text{ cm}) \]
\[ V_{\text{outer}} = \pi (36 \text{ cm}^2) (25 \text{ cm}) \]
\[ V_{\text{outer}} = 900\pi \text{ cm}^3 \]

### Step 5: Calculate the volume of the inner cylinder
Using the same formula for the inner cylinder:
\[ V_{\text{inner}} = \pi r^2 h \]
Substituting the values:
\[ V_{\text{inner}} = \pi (5.2 \text{ cm})^2 (25 \text{ cm}) \]
\[ V_{\text{inner}} = \pi (27.04 \text{ cm}^2) (25 \text{ cm}) \]
\[ V_{\text{inner}} = 676\pi \text{ cm}^3 \]

### Step 6: Calculate the volume of the metal
The volume of the metal is the difference between the outer volume and the inner volume:
\[ V_{\text{metal}} = V_{\text{outer}} - V_{\text{inner}} \]
\[ V_{\text{metal}} = 900\pi \text{ cm}^3 - 676\pi \text{ cm}^3 \]
\[ V_{\text{metal}} = (900 - 676)\pi \text{ cm}^3 \]
\[ V_{\text{metal}} = 224\pi \text{ cm}^3 \]

### Step 7: Approximate the volume
Using \( \pi \approx 3.14 \):
\[ V_{\text{metal}} \approx 224 \times 3.14 \text{ cm}^3 \]
\[ V_{\text{metal}} \approx 703.36 \text{ cm}^3 \]

### Final Answer
The volume of the metal in the cylindrical tube is approximately \( 703.36 \text{ cm}^3 \).

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