$\lambda = \frac{v_c}{f} = \frac{3\times10^{10}}{10\times10^9} = 3cm \ \lambda_c = \frac{2\pi r}{(kr)} = \frac{2 \pi \times 4/2}{1.84} = \frac{4\pi}{1.84} = 6.83 cm $ $ \ \lambda_p = \frac{\lambda}{\sqrt{1-(\lambda/\lambda_0)^2}} = \frac{3}{\sqrt{1-(3/6.83)^2}} = \frac{3}{\sqrt{1-0.193}} = 3.34 cm \ Z_0 = \frac{x}{\sqrt{1-(\frac{\lambda}{\lambda_0})^2}} = \frac{120\pi}{0.898} = 420 \Omega $