If the errors involved in the measurements of a side and mass of a cube are 3% and 4% respectively, what is the maximum permissible error in the density of the material?

The correct option is 13 %.

Step-by-Step Explanation:

Density ($\rho$) of a material is defined as its mass per unit volume:
$\rho =\frac{M}{V}$
where $M$ is the mass of the cube and $V$ is its volume.

For a cube of side length $a$, the volume $V$ is given by:
$V=a^3$

Substituting this volume formula into the density expression gives:
$\rho =\frac{M}{a^3}$

To find the maximum permissible relative error in the density, we take the fractional error of both sides. In error propagation, fractional errors add up, and any exponent becomes a multiplier:
$\frac{\Delta \rho }{\rho }=\frac{\Delta M}{M}+3·\frac{\Delta a}{a}$

Multiplying the entire equation by 100 yields the percentage error relation:
$\left(\frac{\Delta \rho }{\rho }\times 100\right)=\left(\frac{\Delta M}{M}\times 100\right)+3·\left(\frac{\Delta a}{a}\times 100\right)$

From the given data, we have:
- Percentage error in mass ($\frac{\Delta M}{M}\times 100$) = 4%
- Percentage error in side length ($\frac{\Delta a}{a}\times 100$) = 3%

Substitute these values into the percentage error formula:
$\text{Maximum percentage error in density}=4\%+3·(3\%)$

Calculate the final percentage error:
$\text{Maximum percentage error in density}=4\%+9\%=13\%$

Therefore, the maximum permissible error in the density of the material is 13%.