To solve the problem of finding the density of a metal cube and its percentage error, we can follow these steps:
### Step 1: Identify the given values
- Mass of the cube, \( m = 10 \, \text{kg} \) with an uncertainty of \( \Delta m = 0.01 \, \text{kg} \)
- Length of one side of the cube, \( L = 1 \, \text{m} \) with an uncertainty of \( \Delta L = 0.02 \, \text{m} \)
### Step 2: Calculate the volume of the cube
The volume \( V \) of a cube is given by the formula:
\[
V = L^3
\]
Substituting the value of \( L \):
\[
V = (1 \, \text{m})^3 = 1 \, \text{m}^3
\]
### Step 3: Calculate the density
Density \( D \) is defined as mass divided by volume:
\[
D = \frac{m}{V}
\]
Substituting the values:
\[
D = \frac{10 \, \text{kg}}{1 \, \text{m}^3} = 10 \, \text{kg/m}^3
\]
### Step 4: Calculate the percentage error in density
The formula for the percentage error in density is:
\[
\frac{\Delta D}{D} \times 100 = \frac{\Delta m}{m} + 3 \frac{\Delta L}{L}
\]
### Step 5: Calculate the individual errors
1. Calculate \( \frac{\Delta m}{m} \):
\[
\frac{\Delta m}{m} = \frac{0.01 \, \text{kg}}{10 \, \text{kg}} = 0.001
\]
2. Calculate \( \frac{\Delta L}{L} \):
\[
\frac{\Delta L}{L} = \frac{0.02 \, \text{m}}{1 \, \text{m}} = 0.02
\]
### Step 6: Substitute into the percentage error formula
Now substituting these values into the percentage error formula:
\[
\frac{\Delta D}{D} \times 100 = (0.001) + 3(0.02)
\]
Calculating:
\[
\frac{\Delta D}{D} \times 100 = 0.001 + 0.06 = 0.061
\]
### Step 7: Final percentage error
Now, multiplying by 100 to convert to percentage:
\[
\Delta D = 0.061 \times 100 = 6.1\%
\]
### Step 8: Present the final result
Thus, the density of the metal cube is:
\[
D = 10 \, \text{kg/m}^3 \pm 6.1\%
\]
