Question

The density of a solid metal sphere is determined by measuring its mass and its diameter. The maximum error in the density of the sphere is $\left(\frac{x}{100}\right) \% .$ If the relative errors in measuring the mass and the diameter are $6.0 \%$ and $1.5 \%$ respectively, the value of $x$ is.

Answer 1

$\rho=\frac{ M }{ V }=\frac{ M }{\frac{4}{3} \pi\left(\frac{ D }{2}\right)^{3}}$
$\rho=\frac{6}{\pi} M D ^{-3}$
taking log
$\ell n \rho=\ell n \left(\frac{6}{\pi}\right)+\ell nM -3 \ell mD$
Differentiates
$\frac{ d \rho}{\rho}=0+\frac{ dM }{ M }-3 \frac{ d ( D )}{ D }$
for maximum erron
$100 \times \frac{ d \rho}{\rho}=\frac{ d M }{ M } \times 100+\frac{3 dD }{ D } \times 100$
$=6+3 \times 1.5$
$=10.5 \%$
$=\frac{1050}{100} \%$ so $x=1050.00$