The angle of elevation theta of …

From point O angle of elevation is {tex}\theta{/tex} and from point P it is {tex}\phi{/tex}, OP =240 m.

Let PB = x m.
{tex}\tan \theta = \frac { 5 } { 12 } ; \tan \phi = \frac { 3 } { 4 }{/tex}
In right angled {tex}\triangle{/tex}OBA,
{tex}\frac { \mathrm { AB } } { \mathrm { OB } } = \tan \theta{/tex}
{tex}\Rightarrow \frac { h } { 240 + x } = \frac { 5 } { 12 }{/tex} .....(i)
 In right-angled {tex}\triangle{/tex}PBA,
{tex}\frac { A B } { P B } = \tan \phi{/tex}
{tex}\Rightarrow \quad \frac { h } { x } = \frac { 3 } { 4 }{/tex} .......(ii)
Dividing (i) by (ii), we get
{tex}\frac { h } { 240 + x } \times \frac { x } { h } = \frac { 5 } { 12 } \times \frac { 4 } { 3 }{/tex}
{tex}\Rightarrow \frac { x } { 240 + x } = \frac { 5 } { 9 } \Rightarrow 9 x = 1200 + 5 x{/tex}
{tex}\Rightarrow \quad 4 x = 1200 \Rightarrow x = 300{/tex}
Putting x= 300 in (ii) we get, {tex}h = \frac { 3 } { 4 } \times 300 = 225 \mathrm { m }{/tex}
Hence, height of lighthouse is 225 metres.