\[ \text{Height of lighthouse } = h = 100\ \text{m} \] \[ \tan 30^\circ = \frac{h}{d_1} \;\Rightarrow\; d_1 = \frac{h}{\tan 30^\circ} = \frac{100}{1/\sqrt{3}} = 100\sqrt{3}\ \text{m} \] \[ \tan 45^\circ = \frac{h}{d_2} \;\Rightarrow\; d_2 = \frac{100}{1} = 100\ \text{m} \] \[ \text{Ships are on opposite sides} \;\Rightarrow\; \text{distance between ships} = d_1 + d_2 = 100\sqrt{3} + 100 = 100(\sqrt{3}+1)\ \text{m} \] \[ \boxed{\,100(\sqrt{3}+1)\ \text{m} \;\approx\; 273.2\ \text{m}\,} \]