A man standing south of a lamp post, observes his shadow on the horizontal plane to be 24 ft. long, on walking 300 ft. east wards, he finds his shadow as 30 ft. His height is 6 ft.
find the height of the lamp post in ft

Let OP be the lamp post of height h, AB be the first position of the man with shadow $AC=24ft$ and A'B' be the second position of the man with shadow $A^{\prime }{C}^{\prime }=30ft$
Then $AB=A^{\prime }{B}^{\prime }=6$ and $A{A}^{\prime }=300ft$
Let $OA=x$
Now triangle POC and BAC are similar
Therefore $$\begin{gathered} \frac{OP}{AB}=\frac{OC}{AC}\Rightarrow \frac{h}{6}=\frac{x+24}{24} \\ \Rightarrow 4h-24=x \end{gathered}$$
Similarly triangle POC' and B'A'C' are similar 
Therefore $$\begin{gathered} \frac{OP}{A^{\prime }{B}^{\prime }}=\frac{O{C}^{\prime }}{A^{\prime }{C}^{\prime }}\Rightarrow \frac{h}{6}=\frac{O{A}^{\prime }+30}{30} \\ \Rightarrow 5h=\sqrt{x^2+{300}^2}+30 \end{gathered}$$
Now from right angle triangle $△OA{A}^{\prime }$
$$\begin{gathered} \Rightarrow {\left(5h-30\right)}^2={\left(4h-24\right)}^2+{300}^2 \\ \Rightarrow 9h^2-108h-89676=0 \\ \Rightarrow h^2-12h-9964=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow h=\frac{12\pm \sqrt{144+39856}}{2}=6\pm 100 \\ \Rightarrow h=106 \end{gathered}$$