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 length are 4% and 3% respectively, the maximum error in the measurement of density will be

Options

A. 0.07
B. 0.09
C. 0.12
D. 0.13

Explanation

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

### Step 1: Understand the formula for density
Density (ρ) is defined as mass (m) per unit volume (V). For a cube, the volume (V) is given by the cube of the length of its side (L):
\[ \text{Density} (\rho) = \frac{m}{V} = \frac{m}{L^3} \]

### Step 2: Identify the errors in measurements
We are given:
- Maximum error in mass (Δm/m) = 4% = 0.04
- Maximum error in length (ΔL/L) = 3% = 0.03

### Step 3: Use the formula for maximum error in density
The formula for the maximum relative error in density (Δρ/ρ) considering the errors in mass and length is:
\[ \frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3 \cdot \frac{\Delta L}{L} \]

### Step 4: Substitute the values into the formula
Now, substituting the values we have:
\[ \frac{\Delta \rho}{\rho} = 0.04 + 3 \cdot 0.03 \]

### Step 5: Calculate the maximum error in density
Calculating the right side:
\[ \frac{\Delta \rho}{\rho} = 0.04 + 0.09 = 0.13 \]

### Step 6: Convert to percentage
To express this as a percentage, we multiply by 100:
\[ \Delta \rho \text{ (in %)} = 0.13 \times 100 = 13\% \]

### Conclusion
The maximum error in the measurement of density is **13%**.

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