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 ?

To find the maximum permissible error in the density of the material, we start by expressing the density of a cube in terms of its mass and side length.

The formula for density ($\rho$) is given by:
$\rho =\frac{M}{V}$
where:
$M$ is the mass of the cube, and
$V$ is the volume of the cube.

Since the volume of a cube with side length $a$ is $V=a^3$, we can rewrite the density formula as:
$\rho =\frac{M}{a^3}$

To calculate the fractional or relative error, we take the natural logarithm of both sides and then differentiate, or apply the standard rules of error propagation. For a relation of the form $x=\frac{y}{z^n}$, the maximum relative error is:
$\frac{\Delta x}{x}=\frac{\Delta y}{y}+n\frac{\Delta z}{z}$

Applying this rule to our density equation, the maximum relative error in density is:
$\frac{\Delta \rho }{\rho }=\frac{\Delta M}{M}+3\frac{\Delta a}{a}$

To find the percentage error, we multiply each term by 100:
$\frac{\Delta \rho }{\rho }\times 100=(\frac{\Delta M}{M}\times 100)+3(\frac{\Delta a}{a}\times 100)$

We are given the following percentage errors in the problem:
1. Percentage error in mass ($\frac{\Delta M}{M}\times 100$) = 4%
2. Percentage error in side length ($\frac{\Delta a}{a}\times 100$) = 3%

Substituting these values into our error propagation equation:
$\text{Maximum percentage error in density}=4\%+3\times 3\%$
$\text{Maximum percentage error in density}=4\%+9\%=13\%$

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