The measured mass and volume of a body are 53.63 g and `5.8 cm^(3)` respectively, with possible errors of 0.01 g and 0.1 `cm^(3)`. The maximum percentage error in density is about

Options

A. `0.2%`
B. `2%`
C. `5%`
D. `10%`

Answer

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

### Step 1: Understand the formula for density
Density (ρ) is defined as mass (m) divided by volume (V):
\[ \rho = \frac{m}{V} \]

### Step 2: Identify the given values
- Measured mass (m) = 53.63 g
- Measured volume (V) = 5.8 cm³
- Possible error in mass (Δm) = 0.01 g
- Possible error in volume (ΔV) = 0.1 cm³

### Step 3: Calculate the density
Using the measured values:
\[ \rho = \frac{53.63 \, \text{g}}{5.8 \, \text{cm}^3} \approx 9.24 \, \text{g/cm}^3 \]

### Step 4: Determine the relative errors
The relative error in density can be calculated using the formula for propagation of errors:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + \frac{\Delta V}{V} \]
Where:
- \(\Delta \rho\) = error in density
- \(\Delta m\) = error in mass
- \(\Delta V\) = error in volume

### Step 5: Calculate the relative errors for mass and volume
1. **Relative error in mass**:
\[ \frac{\Delta m}{m} = \frac{0.01 \, \text{g}}{53.63 \, \text{g}} \approx 0.000186 \]

2. **Relative error in volume**:
\[ \frac{\Delta V}{V} = \frac{0.1 \, \text{cm}^3}{5.8 \, \text{cm}^3} \approx 0.017241 \]

### Step 6: Combine the relative errors
Now, we can add the relative errors:
\[ \frac{\Delta \rho}{\rho} = 0.000186 + 0.017241 \approx 0.017427 \]

### Step 7: Convert to percentage
To find the maximum percentage error in density:
\[ \text{Percentage error} = \left( \frac{\Delta \rho}{\rho} \right) \times 100 \approx 0.017427 \times 100 \approx 1.74\% \]

### Step 8: Round the result
Rounding to the nearest whole number gives us approximately:
\[ \text{Maximum percentage error in density} \approx 2\% \]

### Final Answer
The maximum percentage error in density is about **2%**.

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