A wire has a mass `(0.3+-0.003)g`, radius `(0.5+-0.005)mm` and length `(6+-0.06)cm`. 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 will 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: Express the volume in terms of radius and length
The volume \(V\) of the wire can be expressed as:
\[ V = A \cdot l = \pi r^2 \cdot l \]
where \(A\) is the cross-sectional area and \(l\) is the length.

### Step 3: Write the formula for density in terms of radius and length
Substituting the expression for volume into the density formula, we have:
\[ \rho = \frac{m}{\pi r^2 l} \]

### Step 4: Determine the errors in measurements
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 relative errors
The relative errors for each measurement are calculated as follows:
- For mass:
\[ \frac{\Delta m}{m} = \frac{0.003}{0.3} = 0.01 \quad \text{(or 1\%)} \]
- For radius:
\[ \frac{\Delta r}{r} = \frac{0.005 \times 10^{-3}}{0.5 \times 10^{-3}} = 0.01 \quad \text{(or 1\%)} \]
- For length:
\[ \frac{\Delta l}{l} = \frac{0.06 \times 10^{-2}}{6 \times 10^{-2}} = 0.01 \quad \text{(or 1\%)} \]

### Step 6: Calculate the maximum percentage error in density
The formula for the maximum percentage error in density is given by:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2 \frac{\Delta r}{r} + \frac{\Delta l}{l} \]
Substituting the values we calculated:
\[ \frac{\Delta \rho}{\rho} = 0.01 + 2(0.01) + 0.01 = 0.01 + 0.02 + 0.01 = 0.04 \]
To express this as a percentage:
\[ \text{Percentage error} = 0.04 \times 100 = 4\% \]

### Final Answer
The maximum percentage error in the measurement of its density is \(4\%\).

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