(a) $\lambda_1=500 \mathrm{~nm}=5 \times 10^{-7} \mathrm{~m}$
(b) $\lambda_2=50 \mu \mathrm{m}=5 \times 10^{-5} \mathrm{~m}$
(c) $\lambda_3=0.500 \mathrm{~nm}=5 \times 10^{-10} \mathrm{~m}$
Let a be the slit width.
(a) $\begin{aligned} W & =\frac{y \lambda_1}{a} \\ \therefore a & =\frac{y \lambda_1}{W}=\frac{(2.5)\left(5 \times 10^{-7}\right)}{3 \times 10^{-3}} \\ & =4.167 \times 10^{-4} m \\ & =0.4167 mm \end{aligned}$
(b) $\begin{aligned} W & =\frac{y \lambda_2}{a} \\ \therefore a & =\frac{y \lambda_2}{W}=\frac{(2.5)\left(5 \times 10^{-5}\right)}{3 \times 10^{-3}} \\ & =4.167 \times 10^{-2} m \\ & =41.67 mm \end{aligned}$
(c) $\begin{aligned} W & =\frac{y \lambda_3}{a} \\ \therefore a & =\frac{y \lambda_3}{W}=\frac{(2.5)\left(5 \times 10^{-10}\right)}{3 \times 10^{-3}} \\ & =4.167 \times 10^{-7} m \\ & =4.167 \times 10^{-4} mm \end{aligned}$
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