Error in the measurement of radius of the sph | vedclass

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(C) Step $1$: Recall the formula for the volume of a sphere.$V = \frac{4}{3} \pi r^3$Step $2$: Apply the rule for the propagation of errors.If a quant

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$6$

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(C) Step $1$: Recall the formula for the volume of a sphere.$V = \frac{4}{3} \pi r^3$Step $2$: Apply the rule for the propagation of errors.If a quantity $Q$ depends on $x$ as $Q = k \cdot x^n$,the relative error is given by $\frac{\Delta Q}{Q} = n \cdot \frac{\Delta x}{x}$.Therefore,the percentage error is $\frac{\Delta Q}{Q} 	imes 100 \% = n \cdot \left( \frac{\Delta x}{x} 	imes 100 \% ight)$.Step $3$: Apply this to the volume formula.Since $V = \frac{4}{3} \pi r^3$,the volume $V$ is proportional to $r^3$ $(V \propto r^3)$.Thus,the percentage error in $V$ is $\Delta V \% = 3 \cdot \Delta r \%$.Step $4$: Substitute the given value.Given $\Delta r \% = 2 \%$.Therefore,$\Delta V \% = 3 	imes 2 \% = 6 \%$.

Error in the measurement of radius of the sphere is $2 \%$. The error in the calculated value of its volume is (in $\%$)

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