A certain body weighs `22.42 g` and has a measured volumen of `4.7 c c`. The possible error in the measurement of mass and volumen are `0.01g` and `0.1 c c`. Then, maximum error in the density will be

Options

A. `22%`
B. `2%`
C. `0.2%`
D. `0.02%`

Explanation

To find the maximum error in the density of a body given its mass and volume along with their respective errors, we can follow these steps:

### Step 1: Understand the formula for density
Density (D) is defined as the mass (M) of an object divided by its volume (V):
\[ D = \frac{M}{V} \]

### Step 2: Identify the given values
- Mass (M) = 22.42 g
- Volume (V) = 4.7 cm³
- Error in mass (ΔM) = 0.01 g
- Error in volume (ΔV) = 0.1 cm³

### Step 3: Calculate the density
Using the formula for density:
\[ D = \frac{M}{V} = \frac{22.42 \, \text{g}}{4.7 \, \text{cm}^3} \]

Calculating this gives:
\[ D \approx 4.77 \, \text{g/cm}^3 \]

### Step 4: Calculate relative errors
The relative error in density can be calculated using the formula:
\[ \frac{\Delta D}{D} = \frac{\Delta M}{M} + \frac{\Delta V}{V} \]

Where:
- \(\Delta D\) = maximum error in density
- \(\Delta M\) = error in mass
- \(\Delta V\) = error in volume

### Step 5: Calculate the individual relative errors
1. Calculate the relative error in mass:
\[ \frac{\Delta M}{M} = \frac{0.01 \, \text{g}}{22.42 \, \text{g}} \approx 0.000445 \]

2. Calculate the relative error in volume:
\[ \frac{\Delta V}{V} = \frac{0.1 \, \text{cm}^3}{4.7 \, \text{cm}^3} \approx 0.02128 \]

### Step 6: Combine the relative errors
Now, we can add the relative errors:
\[ \frac{\Delta D}{D} = 0.000445 + 0.02128 \approx 0.021725 \]

### Step 7: Calculate the maximum error in density
Now, we can find the maximum error in density:
\[ \Delta D = D \times \frac{\Delta D}{D} \]
\[ \Delta D = 4.77 \, \text{g/cm}^3 \times 0.021725 \approx 0.1037 \, \text{g/cm}^3 \]

### Final Result
The maximum error in the density is approximately:
\[ \Delta D \approx 0.10 \, \text{g/cm}^3 \]

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