Correct Answer - Option 2 :
\((18\pm 0.29)kg\)
Correct option-2
Concept:
Multiplication operation in error-
\(\text{Let a quantity }x=ab \)
\( \text{Then, }(x\pm \Delta x)=(a\pm \Delta a)(b\pm \Delta b)\)
\( \Rightarrow x\left( 1\pm \frac{\Delta x}{x} \right)=ab\left( 1\pm \frac{\Delta a}{a} \right)\left( 1\pm \frac{\Delta b}{b} \right)\)
\( \Rightarrow \left( 1\pm \frac{\Delta x}{x} \right)=\left( 1\pm \frac{\Delta a}{a} \right)\left( 1\pm \frac{\Delta b}{b} \right) \)
\(\Rightarrow 1\pm \frac{\Delta x}{x}=1\pm \frac{\Delta a}{a}\pm \frac{\Delta b}{b}\pm \frac{\Delta a}{a}\centerdot \frac{\Delta b}{b}\)
\(Here,\frac{\Delta a}{a}\centerdot \frac{\Delta b}{b}\text{ is very small,so neglected}\text{.} \)
\( \Rightarrow \pm \frac{\Delta x}{x}=\pm \frac{\Delta a}{a}\pm \frac{\Delta b}{b}\)
\(\rm {Hence, maximum \ possible \ value \ of }\)
\( \frac{\Delta x}{x}=\pm \left( \frac{\Delta a}{a}+\frac{\Delta b}{b} \right)\)
Calculation:
Given :-
Density, \(d=\left ( 9\pm 0.2 \right )kg/m^{3}\)
Volume, \(V=\left ( 2\pm 0.01 \right )m^{3}\)
Since, Mass = density * volume
\(\Rightarrow m=d\times V\)
\(\Rightarrow m=\left ( 9\times 2 \right )=18kg\)
Now, \(\frac{\Delta m}{m}=\pm \left ( \frac{\Delta d}{d} +\frac{\Delta v}{v}\right )\)
\(\Rightarrow \Delta m=\pm \left [ \left ( \frac{0.01}{2} +\frac{0.2}{9}\right )\times 18\right ]kg\)
\(\Rightarrow \Delta m=\pm \left ( 0.09+0.2 \right )kg\)
\(\Rightarrow \Delta m=\pm \left ( 0.09+0.2 \right )kg\)
\(\Rightarrow m=\left ( m\pm \Delta m \right )\)
\(\therefore m=\left ( 18\pm 0.29 \right )kg\)
Hence, option-2 is correct
