Trigonometry: Lighthouse and Shadows

Using similar triangles $\Delta KLF \sim \Delta DEF$ and $\Delta KLC \sim \Delta ABC$, we get $LC/LF = 24/30 = 4/5$. With $LB^2 + BE^2 = LE^2$ where $BE = 300$, solving $25x^2 - 16x^2 = 90000 \implies x = 100 \implies LE = 500, LB = 400$. $LC = 424$. $KL = (424/24) \times 6 = 106$ meters.

Horizontal distance from lighthouse to man in second position $LE = 500$ meters.