A physical quantity W is defined by the relation W = A^2 B^3 C ....

The given relation is $W = \dfrac{A^2 B^3}{\sqrt{C}}$.

Let the percentage errors in the measurement of $A$, $B$, and $C$ be $x$, $2x$, and $4x$ respectively, as their ratio is $1 : 2 : 4$.

The maximum percentage error in $W$ is given by:

\left(\dfrac{\Delta W}{W}\right) \times 100 = 2\left(\dfrac{\Delta A}{A} \times 100\right) + 3\left(\dfrac{\Delta B}{B} \times 100\right) + \dfrac{1}{2}\left(\dfrac{\Delta C}{C} \times 100\right)$<br><br>Substituting the assumed percentage errors:<br><br>\left(\dfrac{\Delta W}{W}\right) \times 100 = 2(x) + 3(2x) + \dfrac{1}{2}(4x)$

\left(\dfrac{\Delta W}{W}\right) \times 100 = 2x + 6x + 2x = 10x$<br><br>We are given that the total maximum percentage error in$W$is$30\%$.<br><br>$\Rightarrow 10x = 30\% \Rightarrow x = 3\%$<br><br>The percentage error in the measurement of quantity$B$is$2x = 2(3\%) = 6\%$.<br><br>Answer:$6\%