A wire has a mass `0.3+-0.003g`, radius `0.5+-0.005mm` and length `6+-0.06cm`. The maximum percentage error in the measurement of its density is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the maximum percentage error in the measurement of the density of the wire, we can follow these steps:

### Step 1: Understand the formula for density
The density (ρ) of the wire is given by the formula:
\[ \rho = \frac{m}{V} \]
where \(m\) is the mass and \(V\) is the volume of the wire.

### Step 2: Determine the volume of the wire
The volume \(V\) of the wire can be calculated using the formula for the volume of a cylinder:
\[ V = \pi r^2 L \]
where \(r\) is the radius and \(L\) is the length of the wire.

### Step 3: Calculate the maximum percentage error in density
The maximum percentage error in density can be calculated using the formula:
\[ \frac{\Delta \rho}{\rho} \times 100 \]
where \(\Delta \rho\) is the maximum error in density.

### Step 4: Determine the errors in mass, radius, and length
Given:
- Mass \(m = 0.3 \pm 0.003 \, \text{g}\)
- Radius \(r = 0.5 \pm 0.005 \, \text{mm} = 0.5 \times 10^{-3} \pm 0.005 \times 10^{-3} \, \text{m}\)
- Length \(L = 6 \pm 0.06 \, \text{cm} = 0.06 \pm 0.0006 \, \text{m}\)

### Step 5: Calculate the percentage errors
1. **Mass error**:
\[ \text{Percentage error in mass} = \frac{\Delta m}{m} \times 100 = \frac{0.003}{0.3} \times 100 = 1\% \]

2. **Radius error**:
\[ \text{Percentage error in radius} = \frac{\Delta r}{r} \times 100 = \frac{0.005 \times 10^{-3}}{0.5 \times 10^{-3}} \times 100 = 1\% \]
Since the volume depends on \(r^2\), we multiply this error by 2:
\[ \text{Percentage error in volume due to radius} = 2 \times 1\% = 2\% \]

3. **Length error**:
\[ \text{Percentage error in length} = \frac{\Delta L}{L} \times 100 = \frac{0.06}{6} \times 100 = 1\% \]

### Step 6: Combine the errors
The total percentage error in density is the sum of the individual percentage errors:
\[ \text{Total percentage error in density} = \text{Percentage error in mass} + \text{Percentage error in volume due to radius} + \text{Percentage error in length} \]
\[ = 1\% + 2\% + 1\% = 4\% \]

### Conclusion
The maximum percentage error in the measurement of the density of the wire is **4%**.

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