For $n^{th}$ minima:

$\mathit{\Rightarrow }b\sin \theta =n\lambda$

($\lambda$ is very small, so $\sin \theta$ is very small, hence $\sin \theta \simeq \tan \theta$.

$\Rightarrow b\tan \theta =n\lambda$

$\Rightarrow b\frac{\mathit{y}}{\mathit{D}}\mathit{=}n\lambda$

$\Rightarrow y_n=\frac{n\lambda D}{b}$ (Position of $n^{th}$ minima)

$B\to 1^{\text{st }}$ minima and $A\to 3^{\text{rd }}$:

$y_3=\frac{3\lambda D}{b},y_1=\frac{\lambda D}{b}$

$\Rightarrow \Delta y=y_3-y_1=\frac{2\lambda D}{b}$

$\Rightarrow 3\times {10}^{-3}=\frac{2\times 6000\times {10}^{-10}\times 0.5}{b}$

$\Rightarrow b=\frac{2\times 6000\times {10}^{-10}\times 0.5}{3\times {10}^{-3}}$

$\Rightarrow b=2\times {10}^{-4} \mathrm{m}$