Lets say length of platform is $\text{L metre}.$
We know that $\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}$
$S_{\text{train}} = \dfrac{300}{18}\;\text{metre/sec} = \dfrac{50}{3}\;\text{metre/sec} $
$S_{\text{train}} = \dfrac{300 + L}{39}\;\text{metre/sec}$
$ \implies \dfrac{50}{3} = \dfrac{300 + L}{39}$
$\implies 300 + L = 50 \times 13 = 650$
$\implies L = 350\;\text{metre}.$
So, the correct answer is $(B).$