A thin wire has length of 21.7 cm and radius 0.46 mm. Calculate the volume of the wire to correct significant figures?

Explanation

To calculate the volume of the thin wire, we will follow these steps:

### Step 1: Identify the given values
- Length of the wire (L) = 21.7 cm
- Radius of the wire (r) = 0.46 mm

### Step 2: Convert the radius from mm to cm
Since the length is given in centimeters, we need to convert the radius from millimeters to centimeters:
- 1 mm = 0.1 cm
- Therefore, \( r = 0.46 \, \text{mm} \times 0.1 \, \text{cm/mm} = 0.046 \, \text{cm} \)

### Step 3: Use the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 L \]
where \( r \) is the radius and \( L \) is the length.

### Step 4: Substitute the values into the formula
Now, substituting the values of \( r \) and \( L \) into the formula:
\[ V = \pi (0.046 \, \text{cm})^2 (21.7 \, \text{cm}) \]

### Step 5: Calculate \( r^2 \)
First, calculate \( r^2 \):
\[ (0.046 \, \text{cm})^2 = 0.002116 \, \text{cm}^2 \]

### Step 6: Calculate the volume
Now, substitute \( r^2 \) back into the volume formula:
\[ V = \pi \times 0.002116 \, \text{cm}^2 \times 21.7 \, \text{cm} \]
Calculating this gives:
\[ V \approx 3.14159 \times 0.002116 \times 21.7 \approx 0.1443112 \, \text{cm}^3 \]

### Step 7: Round to the correct significant figures
Now, we need to round the volume to the correct number of significant figures. The length (21.7 cm) has 3 significant figures, and the radius (0.46 mm, converted to 0.046 cm) has 2 significant figures. The final answer should be reported with the least number of significant figures, which is 2 in this case:
\[ V \approx 0.14 \, \text{cm}^3 \]

### Final Answer
The volume of the wire is \( 0.14 \, \text{cm}^3 \).

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