an observer standing 70m high from a building notices that the angles of elevation of the top and bottom of a flagstaff are 60 and 45 degree.find the height of the flagstaff.

I think your question is,
An observer standing 70 m away from a building notices that angles of elevation of top and bottom of the flafstaff (surmounted on the building) are 60 and 45.
Find the height of the flagstaff ?

Here is the solution.



$$\begin{gathered} \mathrm{Let} \mathrm{AC} \mathrm{be} \mathrm{the} \mathrm{flagstaff} \mathrm{that} \mathrm{is} \mathrm{surmounted} \mathrm{on} \mathrm{a} \mathrm{building} \mathrm{CB}. \\ \mathrm{Let} \mathrm{the} \mathrm{height} \mathrm{of} \mathrm{the} \mathrm{flagstaff} , \mathrm{AC} = \mathrm{h} \\ \mathrm{Let} \mathrm{the} \mathrm{height} \mathrm{of} \mathrm{the} \mathrm{building}, \mathrm{CB} = \mathrm{H} \\ \mathrm{Let} \mathrm{P} \mathrm{be} \mathrm{the} \mathrm{point} \mathrm{of} \mathrm{observation}. \\ \mathrm{Now}, \mathrm{BP} = 70 \mathrm{m} \\ \mathrm{Let} \angle \mathrm{CPB} = 45°; \angle \mathrm{APB} = 60° \\ \mathrm{In} ∆\mathrm{CBP}, \\ \tan 45° = \frac{\mathrm{CB}}{\mathrm{BP}} \\ \Rightarrow 1 = \frac{\mathrm{H}}{70} \\ \Rightarrow \mathrm{H} = 70 \mathrm{m} \\ \mathrm{In} ∆\mathrm{ABP}, \\ \tan 60° = \frac{\mathrm{AB}}{\mathrm{BP}} \\ \Rightarrow \sqrt{3} = \frac{\mathrm{h} + \mathrm{H}}{70} \\ \Rightarrow \mathrm{h} + \mathrm{H} = 70\sqrt{3} \\ \Rightarrow \mathrm{h} + 70 = 70\sqrt{3} \\ \Rightarrow \mathrm{h} = 70\left(\sqrt{3} - 1\right) \\ \Rightarrow \mathrm{h} = 70\left(1.732-1\right) \\ \Rightarrow \mathrm{h} = 70 \times 0.732 \\ \Rightarrow \boxed{\mathrm{h} = 51.24 \mathrm{m}} \\ \mathrm{So}, \mathrm{height} \mathrm{of} \mathrm{flagstaff}, \mathrm{h} = 51.24 \mathrm{m} \end{gathered}$$