What is the maximum percentage error in density of body if percentage error in measurement of mass is `4%` , and percentage error in measurement of side `1%` ?

Options

A. `18%`
B. `7%`
C. `20%`
D. `21%`

Answer

To solve the problem of finding the maximum percentage error in the density of a body given the percentage errors in mass and side measurements, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Percentage error in mass (Δm/m) = 4%
- Percentage error in side length (ΔL/L) = 1%

2. **Understand the Formula for Density**:
- Density (D) is defined as:
\[ D = \frac{m}{V} \]
- For a cube with side length L, the volume (V) is:
\[ V = L^3 \]

3. **Express the Density in Terms of Mass and Length**:
- Thus, the density can be expressed as:
\[ D = \frac{m}{L^3} \]

4. **Calculate the Percentage Error in Density**:
- The formula for the percentage error in density (ΔD/D) is given by:
\[ \frac{\Delta D}{D} = \frac{\Delta m}{m} + 3 \cdot \frac{\Delta L}{L} \]
- Here, the factor of 3 comes from the differentiation of volume with respect to length (since volume is proportional to the cube of the length).

5. **Substitute the Given Percentage Errors**:
- Substitute the values:
\[ \frac{\Delta D}{D} = \frac{4}{100} + 3 \cdot \frac{1}{100} \]
- This simplifies to:
\[ \frac{\Delta D}{D} = 0.04 + 3 \cdot 0.01 = 0.04 + 0.03 = 0.07 \]

6. **Convert to Percentage**:
- To express this as a percentage:
\[ \Delta D \text{ (percentage)} = 0.07 \times 100 = 7\% \]

7. **Conclusion**:
- The maximum percentage error in density is **7%**.

### Final Answer:
The maximum percentage error in density of the body is **7%**.

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