The density of a solid metal sphere is determined by measuring it… - Vidyadip

MCQ

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

A
$1000$
B
$1075$
C
$1060$
✓
$1050$

✓

Answer

Correct option: D.

$1050$

d
$\rho=\frac{M}{V}=\frac{M}{\frac{4}{3} \pi\left(\frac{D}{2}\right)^{3}}$

$\rho=\frac{6}{\pi} M D ^{-3}$

taking log

$\ell n \rho=\ell n \left(\frac{6}{\pi}\right)+\ell n M -3 \ell m D$

Differentiates

$\frac{ d p}{\rho}=0+\frac{ d M }{ M }-3 \frac{ d ( D )}{ D }$

for maximum error

$100 \times \frac{ d \rho}{\rho}=\frac{ dM }{ M } \times 100+\frac{3 d D }{ D } \times 100$

$=6+3 \times 1.5$

$=10.5 \%$

$=\frac{1050}{100} \%$

$x=1050.00$

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