The measured mass and volume of a body are `22.42 g` and `4. cm^(3)`, respectively, with possible errors of `0.01 g` and `0.1 cm^(3)`. The maximum error in density is

Options

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

Explanation

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

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

### Step 2: Identify the given values and their possible errors
- Measured mass, \( m = 22.42 \, \text{g} \) with a possible error \( \Delta m = 0.01 \, \text{g} \)
- Measured volume, \( v = 4.0 \, \text{cm}^3 \) with a possible error \( \Delta v = 0.1 \, \text{cm}^3 \)

### Step 3: Use the formula for maximum error in density
The relative error in density can be calculated using the formula:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + \frac{\Delta v}{v} \]
where \( \Delta \rho \) is the maximum error in density.

### Step 4: Calculate the relative errors
1. Calculate the relative error in mass:
\[ \frac{\Delta m}{m} = \frac{0.01}{22.42} \]
Calculate this value:
\[ \frac{\Delta m}{m} \approx 0.000445 \]

2. Calculate the relative error in volume:
\[ \frac{\Delta v}{v} = \frac{0.1}{4.0} \]
Calculate this value:
\[ \frac{\Delta v}{v} = 0.025 \]

### Step 5: Add the relative errors
Now, add the two relative errors:
\[ \frac{\Delta \rho}{\rho} = 0.000445 + 0.025 \approx 0.025445 \]

### Step 6: Calculate the maximum error in density
To find the maximum error in density, multiply the relative error by the calculated density:
1. First, calculate the density:
\[ \rho = \frac{22.42}{4.0} = 5.605 \, \text{g/cm}^3 \]

2. Now, calculate the maximum error in density:
\[ \Delta \rho = \rho \times \frac{\Delta \rho}{\rho} = 5.605 \times 0.025445 \approx 0.1425 \, \text{g/cm}^3 \]

### Conclusion
The maximum error in density is approximately \( 0.1425 \, \text{g/cm}^3 \).

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