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

Answer: 1050.00

$\rho =\frac{M}{V}=\frac{M}{\frac{4}{3}\pi {\left(\frac{D}{2}\right)}^3}$
$\rho =\frac{6}{\pi }M{D}^{-3}$
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
$100\times \frac{d\rho }{\rho }=\frac{dM}{M}\times 100+\frac{3dD}{D}\times 100$
$=6+3\times 1.5$
$=10.5\%$
$=\frac{1050}{100}\%$ so $x=1050.00$