Question

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively $1.5\mathrm{\%}$ and $1\mathrm{\%}$ the maximum error in determining the density is:
A) $2.5\mathrm{\%}$
B) $3.5\mathrm{\%}$
C) $4.5\mathrm{\%}$
D) $6\mathrm{\%}$

Answer 1

Hint: To solve this question we need to use the formulae for relative error of density of a substance. For this we should know the formulae of density of a substance. Also in a cube, all the sides are equal. In the question, the error in length is given which can be used in place of error inside.

Formula Used:
Density of a cube $\left(\rho \right)$ = ${\displaystyle \frac{m}{a^3}}$
Where $\rho$is the density of the cube, $m$is the mass of the substance and $a$is the side length of the substance.
Also Relative error is ${\displaystyle \frac{\mathrm{\Delta }x}{x}}\times 100\mathrm{\%}$
Where $\mathrm{\Delta }x$is the difference in measured value and actual expected value and $x$is the actual expected value.

Complete step by step answer:
In the question the error in mass is given as $1.5\mathrm{\%}$ and the error in length is given as $1\mathrm{\%}$.
We know the formulae for density is
$\rho ={\displaystyle \frac{m}{a^3}}$
Where $\rho$is the density of the cube, $m$is the mass of the substance and $a$is the side length of the substance.
We know that relative error is given by ${\displaystyle \frac{\mathrm{\Delta }x}{x}}\times 100\mathrm{\%}$.
Where $\mathrm{\Delta }x$is the difference in measured value and actual expected value and $x$is the actual expected value.
The relative error in density will be the sum of the relative errors in the mass and length. So relative error in density is
$\Rightarrow {\displaystyle \frac{\mathrm{\Delta }\rho }{\rho }}\times 100\mathrm{\%}={\displaystyle \frac{\mathrm{\Delta }m}{m}}\times 100\mathrm{\%}+3{\displaystyle \frac{\mathrm{\Delta }a}{a}}\times 100\mathrm{\%}$
$\Rightarrow {\displaystyle \frac{\mathrm{\Delta }\rho }{\rho }}\times 100\mathrm{\%}=1.5\mathrm{\%}+3\times 1\mathrm{\%}=4.5\mathrm{\%}$

Hence, the correct option is (C).

Additional Information:
Relative error when used as a measure of precision is the ratio of the absolute error of a measurement to the measurement being taken. In other words, this type of error is relative to the size of the item being measured.

Note: With the help of relative error method, we can determine the magnitude of the absolute error in terms of the actual size of the measurement. This can be used in error measurement even if the actual dimensions are not known. Lesser the value of relative error, more accurate the value is. Relative error is a dimensionless quantity.