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

The correct option is 9%.

Underlying Principle:
Density ($\rho$) is defined as the mass ($M$) of an object divided by its volume ($V$). For a cube of side length $L$, the volume is given by $V=L^3$. Therefore, the relationship for the density of a cube is:

$\rho =\frac{M}{L^3}$

Error Propagation Formulation:
To find the maximum relative error in density, we apply the rules of error propagation. When quantities are multiplied or divided, their relative or percentage errors add up. For power terms, the relative error is multiplied by the power index. Taking the natural logarithm on both sides of the density formula and then differentiating gives:

$\frac{\Delta \rho }{\rho }=\frac{\Delta M}{M}+3\frac{\Delta L}{L}$

Multiplying both sides by 100 gives the relation for the maximum percentage error:

$(\frac{\Delta \rho }{\rho }\times 100)=(\frac{\Delta M}{M}\times 100)+3\times (\frac{\Delta L}{L}\times 100)$

Calculation:
We are given:
- Maximum error in mass measurement = 3%
- Maximum error in length measurement = 2%

Substituting these values into our error propagation equation:

$\text{Maximum \% error in density}=3\%+3\times 2\%$

$\text{Maximum \% error in density}=3\%+6\%=9\%$

Thus, the maximum error in the measurement of density is 9%.