A wire of length `l =6 +- 0.06cm` and radius `r =0.5+-0.005 cm` and mass `m =0. 3+-0.003gm`. Maximum percentage error in density is

Options

A. a. `4`
B. b. `2`
C. c. `1`
D. d. `6.8`

Explanation

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

### Step 1: Understand the formula for density
The density \( \rho \) 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 a cylindrical wire can be calculated using the formula:
\[ V = \pi r^2 l \]
where \( r \) is the radius and \( l \) is the length of the wire.

### Step 3: Calculate the percentage errors
To find the maximum percentage error in density, we need to calculate the percentage errors in mass, radius, and length.

1. **Percentage error in mass (\( m \))**:
\[ \text{Percentage error in mass} = \left( \frac{\Delta m}{m} \right) \times 100 \]
Given \( m = 0.3 \, \text{g} \) and \( \Delta m = 0.003 \, \text{g} \):
\[ \text{Percentage error in mass} = \left( \frac{0.003}{0.3} \right) \times 100 = 1\% \]

2. **Percentage error in radius (\( r \))**:
\[ \text{Percentage error in radius} = \left( \frac{\Delta r}{r} \right) \times 100 \]
Given \( r = 0.5 \, \text{cm} \) and \( \Delta r = 0.005 \, \text{cm} \):
\[ \text{Percentage error in radius} = \left( \frac{0.005}{0.5} \right) \times 100 = 1\% \]

3. **Percentage error in length (\( l \))**:
\[ \text{Percentage error in length} = \left( \frac{\Delta l}{l} \right) \times 100 \]
Given \( l = 6 \, \text{cm} \) and \( \Delta l = 0.06 \, \text{cm} \):
\[ \text{Percentage error in length} = \left( \frac{0.06}{6} \right) \times 100 = 1\% \]

### Step 4: Combine the percentage errors
The formula for the maximum percentage error in density is given by:
\[ \text{Percentage error in density} = \text{Percentage error in mass} + 2 \times \text{Percentage error in radius} + \text{Percentage error in length} \]
Substituting the values we calculated:
\[ \text{Percentage error in density} = 1\% + 2 \times 1\% + 1\% = 1\% + 2\% + 1\% = 4\% \]

### Final Answer
The maximum percentage error in density is:
\[ \boxed{4\%} \]