The density of a cube is measured by measuring its mass and length of its sides. If the maximum error in the measurement of mass and lengths are 3% and 2% respectively, the maximum error in the measurement of density would be

Options

A. 0.12
B. 0.14
C. 0.07
D. 0.09

Answer

To find the maximum error in the measurement of density when the maximum errors in the measurements of mass and length are given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for density**:
The density \(D\) of a cube is defined as:
\[ D = \frac{m}{V} \]
where \(m\) is the mass and \(V\) is the volume. The volume \(V\) of a cube with side length \(L\) is given by:
\[ V = L^3 \]
Therefore, we can rewrite the density as:
\[ D = \frac{m}{L^3} \]

2. **Identify the errors in measurements**:
- The maximum error in the measurement of mass \(m\) is given as \(3\%\).
- The maximum error in the measurement of length \(L\) is given as \(2\%\).

3. **Calculate the percentage error in density**:
The formula for the percentage error in density can be derived from the errors in mass and volume. The volume \(V\) depends on the cube of the length, so we need to account for that in our error calculations.

The percentage error in density \(D\) can be expressed as:
\[ \frac{\Delta D}{D} \times 100\% = \frac{\Delta m}{m} \times 100\% + 3 \times \frac{\Delta L}{L} \times 100\% \]

Here, \(\Delta m/m\) is the relative error in mass and \(\Delta L/L\) is the relative error in length.

4. **Substituting the given errors**:
- The relative error in mass is \(3\%\) or \(0.03\).
- The relative error in length is \(2\%\) or \(0.02\).

Substituting these values into the formula gives:
\[ \frac{\Delta D}{D} \times 100\% = 3\% + 3 \times 2\% \]
\[ = 3\% + 6\% \]
\[ = 9\% \]

5. **Conclusion**:
Therefore, the maximum error in the measurement of density is:
\[ \text{Maximum error in density} = 9\% \]