Question

For a cubical block, error in measurement of sides is ${}_{-}^{+}1\%$ and error in measurement of mass is ${}_{-}^{+}2\%$, then maximum possible error in density is

• A. $1\%$
• B. $5\%$
• C. $3\%$
• D. $7\%$

Answer 1

The correct option is B $5\%$

$\begin{array}{cc}V& =a^3.......\left(i\right)\\ dV& =3a^{2}da\\ \rho & =\frac{m}{v}.....\left(ii\right)\\ \text{taking}& \log \text{in}\left(ii\right)\\ \text{log}\rho =\log m-\ln y& \\ \text{differentiating}& \end{array}$

$\begin{array}{c}\frac{d\rho }{\rho }=\frac{dm}{m}-\frac{dV}{V}\\ \frac{d\rho }{\rho }=\frac{dm}{m}-\frac{3a^{2}da}{a^3}\\ \text{error is always added}\\ \mathrm{100}\times \left(\frac{d\rho }{\rho }\right)=(\frac{dm}{m}+3\frac{da}{a})\times 100\\ \text{\% error in}\rho \end{array}$

$\begin{array}{c}=(2+(3\times 1\left)\right)\%\\ =5\text{percent}\end{array}$