In the density measurement of a cube, the mass and edge length are measured as `(10.00+-0.10)kg` and `(0.10+-0.01)m`, respectively. The erroer in the measurement of density is :

Options

A. `0.01 kg//m^(3)`
B. `0.10 kg//m^(3)`
C. `0.31 kg//m^(3)`
D. `0.07 kg//m^(3)`

Answer

To find the error in the measurement of 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). For a cube, the volume can be expressed as the cube of the edge length (a):
\[ \rho = \frac{m}{V} = \frac{m}{a^3} \]

### Step 2: Identify the given values
We have the following measurements:
- Mass, \( m = 10.00 \, \text{kg} \) with an uncertainty of \( \Delta m = 0.10 \, \text{kg} \)
- Edge length, \( a = 0.10 \, \text{m} \) with an uncertainty of \( \Delta a = 0.01 \, \text{m} \)

### Step 3: Calculate the volume
The volume of the cube can be calculated as:
\[ V = a^3 = (0.10)^3 = 0.001 \, \text{m}^3 \]

### Step 4: Calculate the density
Now we can calculate the density:
\[ \rho = \frac{m}{V} = \frac{10.00 \, \text{kg}}{0.001 \, \text{m}^3} = 10000 \, \text{kg/m}^3 \]

### Step 5: Calculate the relative error in density
To find the error in density, we will use the formula for the propagation of uncertainties. The relative error in density can be expressed as:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3 \frac{\Delta a}{a} \]

### Step 6: Substitute the values into the formula
Now we can substitute the values:
- Relative error in mass:
\[ \frac{\Delta m}{m} = \frac{0.10}{10.00} = 0.01 \]
- Relative error in edge length:
\[ \frac{\Delta a}{a} = \frac{0.01}{0.10} = 0.1 \]
- Therefore, the total relative error in density is:
\[ \frac{\Delta \rho}{\rho} = 0.01 + 3 \times 0.1 = 0.01 + 0.3 = 0.31 \]

### Step 7: Calculate the absolute error in density
Now we can find the absolute error in density:
\[ \Delta \rho = \rho \times \frac{\Delta \rho}{\rho} = 10000 \times 0.31 = 3100 \, \text{kg/m}^3 \]

### Final Result
The error in the measurement of density is:
\[ \Delta \rho = 3100 \, \text{kg/m}^3 \]