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# 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 $1.5\%$ and $1\%$ the maximum error in determining the density is:A) $2.5\%$B) $3.5\%$C) $4.5\%$D) $6\%$

Last updated date: 17th Sep 2024
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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 $\left( \rho \right)$ = $\dfrac{m}{{{a^3}}}$
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 $\dfrac{{\Delta x}}{x} \times 100\%$
Where $\Delta x$is the difference in measured value and actual expected value and $x$is the actual expected value.

In the question the error in mass is given as $1.5\%$ and the error in length is given as $1\%$.
We know the formulae for density is
$\rho = \dfrac{m}{{{a^3}}}$
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 $\dfrac{{\Delta x}}{x} \times 100\%$.
Where $\Delta x$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
$\Rightarrow \dfrac{{\Delta \rho }}{\rho } \times 100\% = \dfrac{{\Delta m}}{m} \times 100\% + 3\dfrac{{\Delta a}}{a} \times 100\%$
$\Rightarrow \dfrac{{\Delta \rho }}{\rho } \times 100\% = 1.5\% + 3 \times 1\% = 4.5\%$

Hence, the correct option is (C).