Given :
The angle of elevation from her position to the top of a tower is 32°.
The angle of elevation again from a point 50 m closer to the tower and finds it to be 52°.
The following figure represents the given situation :
Let the height of the tower h
And the distance between the tower and the second point = x
So, from the larger triangle :
[tex]\begin{gathered} \tan 32=\frac{h}{x+50} \\ \\ h=(x+50)\cdot\tan 32 \end{gathered}[/tex]From the smaller triangle :
[tex]\begin{gathered} \tan 52=\frac{h}{x} \\ \\ h=x\cdot\tan 52 \end{gathered}[/tex]So,
[tex](x+50)\cdot\tan 32=x\tan 52[/tex]solve for x:
[tex]\begin{gathered} x\cdot\tan 32+50\tan 32=x\cdot\tan 52 \\ 50\cdot\tan 32=x\cdot\tan 52-x\cdot\tan 32 \\ 50\cdot\tan 32=x\cdot(\tan 52-\tan 32) \\ \\ x=\frac{50\cdot\tan 32}{\tan 52-\tan 32}\approx47.7 \end{gathered}[/tex]so, the height h will be :
[tex]h=x\cdot\tan 52=47.7\cdot\tan 52\approx61m[/tex]So, the height of the tower = 61 m
